“The reason I’m here tonight is to put people on the spot,” Mr. Paul said Thursday night. “I want people to feel uncomfortable. I want them to have to answer people at home who said, ‘How come you were against President Obama’s deficits and then how come you’re for Republican deficits?’”

The shutdown came on the heels of a three-day closure brought about by Senate Democrats last month. As midnight approached, Mr. Paul did not relent, bemoaning from the Senate floor what he saw as out-of-control government spending and repeatedly rebuffing attempts by his fellow senators to move ahead with a vote.

“I think the country’s worth a debate until 3 in the morning, frankly,” he said.

Senate leaders were left helpless.

“I think it’s irresponsible,” said Senator John Cornyn of Texas, the No. 2 Senate Republican, lamenting what he described as “the act of a single senator who just is trying to make a point but doesn’t really care too much about who he inconveniences.”

Mr. Paul’s ideological opponents were not buying his fiscal rectitude either. Senator Brian Schatz, Democrat of Hawaii, posted on Twitter: “Rand Paul voted for a tax bill that blew a $1.5 trillion hole in the budget. Now he is shutting the government down for three hours because of the debt. The chance to demonstrate fiscal discipline was on the tax vote. Delaying a vote isn’t a profile in courage, it’s a cleanup.”

The Senate finally passed the measure, 71 to 28, shortly before 2 a.m. The House followed suit around 5:30 a.m., voting 240 to 186 for the bill.

Before Mr. Paul waged his assault on the budget deal, trouble was already brewing in the House, where angry opposition from the Republicans’ most ardent conservative members, coupled with Democratic dissenters dismayed that the deal did nothing for young undocumented immigrants, created new tension as the clock ticked toward midnight.

Representative Nancy Pelosi of California, the Democratic leader, told a closed-door meeting of House Democrats that she would oppose the deal, and said that Democrats would have leverage if they held together to demand a debate on immigration legislation. But she suggested that she would not stand in the way of lawmakers who wanted to vote their conscience.

Pressing the issue further, Ms. Pelosi and the next two highest-ranking House Democrats sent a letter to Mr. Ryan noting their desire for the government to remain open and imploring him to make a public statement about the scheduling of a vote on legislation to protect young undocumented immigrants who are now shielded from deportation by the Obama-era Deferred Action for Childhood Arrivals program, or DACA.

“Most of our members believe the budget agreement is a reasonable compromise to address America’s military strength and critical domestic priorities, like fighting the opioid crisis, boosting N.I.H., moving forward to resolve the pension crisis, caring for our veterans, making college more affordable and investing in child care for working families,” they wrote. “We are writing to again reiterate our request that you make a public statement regarding the scheduling of a vote on a DACA bill.”

The run-up to the House vote, when passage was no foregone conclusion, highlighted the divisions within the Democratic caucus over how hard to push on the issue of immigration as Congress prepares to turn its focus to that politically volatile subject.

The text of the deal, stretching more than 600 pages, was released late Wednesday night, revealing provisions large and small that would go far beyond the basic budget numbers. The accord would raise strict spending caps on domestic and military spending in this fiscal year and the next one by about $300 billion in total. It would also lift the federal debt limit until March 2019.

Critically, it would also keep the government funded for another six weeks, giving lawmakers time to put together a long-term spending bill that would stretch through the rest of the fiscal year, which ends Sept. 30. The previous temporary funding measure, which was passed to end the last shutdown, expired at midnight on Thursday.

The deal had been expected to sail through the Senate, and the House had planned to vote on it later Thursday, until Mr. Paul took his stand.

The White House Office of Management and Budget instructed federal agencies to prepare for a possible lapse in funding, a spokeswoman said Thursday night. Even with a technical lapse in government funding, the effect of the shutdown was limited because lawmakers gave final approval to the deal only hours after funding expired.

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As the midnight deadline approached, Senate leaders from both parties nudged Mr. Paul to stop holding up the vote. And his colleagues had little to do but wait.

“It’s just further example of the dysfunction of this place,” said Senator Ron Johnson, Republican of Wisconsin. “It’s ridiculous, isn’t it?”

Senator Johnny Isakson, Republican of Georgia, offered a succinct account of his evening: “Living the dream.”

Among the Democratic ranks in the House, the objections were also strenuous, but for reasons very different from Mr. Paul’s.

With the monthslong budget impasse appearing to be on the cusp of a resolution, lawmakers were girding for a fight over the fate of young immigrants who were brought to the country illegally as children, known as Dreamers, as well as Mr. Trump’s plan to build a wall along the southern border with Mexico and other possible immigration policy changes.

The uncertain outlook for immigration legislation, and the disagreements on the best strategy to move forward, was starkly apparent as Ms. Pelosi commanded the House floor for more than eight hours on Wednesday in an effort to help the young immigrants. She said she would oppose the budget deal unless Mr. Ryan offered a commitment to hold a vote on legislation in the House that would address the fate of the Dreamers.

On Thursday, Ms. Pelosi herself displayed the conflicting pressures on Democrats. She simultaneously hailed the budget deal while proclaiming she would vote against it. In a letter to colleagues, she explained her opposition to the deal, but also nodded to its virtues and held back from pressuring other Democrats to vote against it.

“I’m pleased with the product,” she told reporters. “I’m not pleased with the process.”

In his own comments to reporters on Thursday, Mr. Ryan stressed his desire to address the fate of the young immigrants. But he did not offer the kind of open-ended commitment that might assuage Ms. Pelosi. Instead, he signaled that whatever bill the House considers would be one that Mr. Trump supports.

“To anyone who doubts my intention to solve this problem and bring up a DACA and immigration reform bill, do not,” he said. “We will bring a solution to the floor, one that the president will sign.”

Just before the vote on Friday morning, Mr. Ryan offered a further reassurance about his commitment to addressing DACA. Once the budget deal has been approved, he said, “we will focus on bringing that debate to this floor and finding a solution.”

The fate of the Dreamers has been in question since Mr. Trump moved in September to end DACA. The president gave Congress six months to come up with a solution to resolve their fate.

In recent months, Democrats have tried to make use of the leverage they have in fiscal negotiations, and the issue of immigration played a central role in last month’s shutdown. But Democrats have struggled to determine how hard they should push.

In last month’s closure, the vast majority of Senate Democrats voted to block a bill that would have kept the government open, only to retreat a few days later and agree to end the closure after Senator Mitch McConnell of Kentucky, the majority leader, promised a Senate debate on immigration.

This time, House Democrats were clearly split in their calculations about the best way to exert influence over immigration.

Representative Luis V. Gutiérrez, Democrat of Illinois, demanded that Ms. Pelosi use her muscle to “stop the Democrats from folding.”

“Anyone who votes for the Senate budget deal is colluding with this president and this administration to deport Dreamers,” he said. “It is as simple as that.”

Democrats also ran the risk of angering liberal activists who want to see them take a stand. Ben Wikler, the Washington director for MoveOn.org, said House Democrats would be making a strategic mistake by voting for the budget deal.

“If you’re looking at a boulder and you have a choice between a lever or your bare hands, you should use the lever,” he said.

But Democrats secured important victories in the budget pact, obtaining big increases in funding for domestic programs. Voting against those wins to take a stand on DACA — and possibly prolonging the shutdown — carried its own political risks.

Representative John Yarmuth of Kentucky, the top Democrat on the House Budget Committee, noted that the budget deal “meets nearly every one of our priorities.”

“If Democrats cannot support this kind of compromise, Congress will never function,” he said.

The spotlight was on House Democrats in part because it had become apparent that Republican leaders would most likely lack the votes to push the budget deal through the House with only votes from their own party.

A sizable number of House Republicans rebelled against the deal because of its huge increase in spending. The conservative House Freedom Caucus, which has roughly three dozen members, formally opposed the deal.

“It was pretty much a smorgasbord of spending and policy that got added to this,” said Representative Mark Meadows, Republican of North Carolina and the chairman of the Freedom Caucus. “Normally, people who eat at smorgasbords all the time are not the healthiest.”

Republicans were similarly upbeat. “I’m optimistic that very soon we’ll be able to reach an agreement,” said Senator Mitch McConnell of Kentucky, the majority leader.

But if the bipartisan deal falls through, lawmakers have no clear plan to keep the government open past Thursday, with the parties disagreeing over spending priorities and the president lobbing verbal bombs from the White House.

“I would shut it down over this issue,” Mr. Trump said as he demanded an immigration deal on his terms. “If we don’t straighten out our border, we don’t have a country.”

Mr. Trump’s call for a shutdown was not the first time he had brandished the threat of closing down the government. He mused on Twitter last year that the country “needs a good ‘shutdown,’” a suggestion Democrats did not forget.

And on Tuesday, in a striking moment, a lawmaker from the president’s own party, Representative Barbara Comstock of Virginia, pushed back at the White House meeting.

“We don’t need a government shutdown on this,” said Ms. Comstock, a top Democratic target in the midterm elections. She represents a moderate district in Northern Virginia, an area that is home to many federal workers.

Soon after, Mr. Trump interrupted her. “You can say what you want,” he said. “We’re not getting support from the Democrats on this legislation.”

It has been less than three weeks since the last shutdown — a three-day closing that ended after Democrats won a promise from Mr. McConnell to have the Senate consider legislation to protect young immigrants who were brought to the United States illegally as children.

Mr. McConnell pledged that the Senate would turn to immigration if no deal had been reached on that subject by Thursday, when the current government funding measure is set to expire. This time, Senate Democrats have shown no appetite to force a shutdown.

But the fate of the young immigrants, known as Dreamers, remains highly uncertain, especially given Mr. Trump’s insistence that Democrats agree to build a wall on the Mexican border and enact other tough immigration policies.

The White House chief of staff, John F. Kelly, also took a hard line on immigration during a visit to the Capitol on Tuesday.

He said Mr. Trump was unlikely to extend a March 5 deadline, when the Obama-era program to protect Dreamers — Deferred Action for Childhood Arrivals, or DACA — is set to expire. And he said the president had been generous in his offer to give 1.8 million young undocumented immigrants a path to citizenship in exchange for a series of hard-line immigration policy changes.

“There are 690,000 official DACA registrants, and the president sent over what amounts to be two and a half times that number, to 1.8 million,” Mr. Kelly told reporters. “The difference between 690 and 1.8 million were the people that some would say were too afraid to sign up, others would say were too lazy to get off their asses, but they didn’t sign up.”

Neither party seemed to have any idea how the immigration debate would play out in the Senate in the days to come. Mr. McConnell has promised a free and open debate, with senators allowed to offer amendments to whatever measure is brought to the floor. But what the starting bill would look like remained a mystery.

“That’s the $64,000 question,” said Senator John Cornyn of Texas, the No. 2 Republican in the Senate. “Everybody wants to know, and Senator McConnell hasn’t told us.”

Mr. McConnell gave little hint of how the debate would proceed.

“In the Senate, on those rare occasions when we have these kind of open debates, whoever gets to 60 wins,” he told reporters. “And it’ll be an opportunity for 1,000 flowers to bloom.”

But with less than 72 hours remaining to avert a shutdown, congressional Republicans were moving on an entirely different track from the president, and immigration was not part of the equation as the House moved ahead with the stopgap spending bill.

The House voted 245 to 182 to approve the bill, with most Democrats voting against it.

House Republicans were hoping to pressure Senate Democrats to go along or face harsh political consequences. The bill would keep the government open through March 23.

“Quite literally the safety of our service members and the security of our country is at stake,” Speaker Paul D. Ryan of Wisconsin said.

But 44 members of the Senate Democratic caucus signed a letter in December opposing the House Republican approach. Passing the bill in the Senate will require 60 votes, meaning that those 44 Democrats, if they stick together, could block it.

“This is not a serious bill,” said Representative Nita M. Lowey of New York, the top Democrat on the House Appropriations Committee. “It is nothing more than a political ploy that will place us on the brink of another shutdown.”

If a deal on the spending caps is reached, lawmakers could approve legislation that includes that agreement along with a temporary measure to keep the government open. The package could also end up carrying other items as well, including disaster aid in response to last year’s hurricanes and perhaps an increase to the statutory limit on the government’s borrowing authority.