WASHINGTON — Veering away from a default crisis, the House overwhelmingly approved a debt ceiling and budget cuts package, sending the deal that President Biden and Speaker McCarthy negotiated to the Senate for swift passage in a matter of days, before a fast-approaching deadline.

The hard-fought compromise pleased few, but lawmakers assessed it was better than the alternative — a devastating economic upheaval if Congress failed to act. Tensions ran high as hard-right Republicans refused the deal, but Messrs. Biden and McCarthy assembled a bipartisan coalition to push to passage on a robust 314-117 vote late Wednesday.

“We did pretty dang good,” Mr. McCarthy said afterward.

Amid deep discontent from Republicans who said the spending restrictions did not go far enough, Mr. McCarthy said it is only a “first step.”

Mr. Biden, watching the tally from Colorado Springs where Thursday he is scheduled to deliver the commencement address at the U.S. Air Force Academy, phoned Mr. McCarthy and the other congressional leaders after the vote. In a statement, he called the outcome “good news for the American people and the American economy.”

Washington is rushing after a long slog of debate to wrap up work on the package to ensure the government can keep paying its bills, and prevent financial upheaval at home and abroad. Next Monday is when the Treasury has said the U.S. would run short of money and risk a dangerous default.

Mr. Biden had been calling lawmakers directly to shore up backing. Mr. McCarthy worked to sell skeptical fellow Republicans, even fending off challenges to his leadership.

A similar bipartisan effort from Democrats and Republicans will be needed in the Senate to overcome objections.

Overall, the 99-page bill would make some inroads in curbing the nation’s deficits as Republicans demanded, without rolling back Trump-era tax breaks as Mr. Biden wanted. To pass it, Messrs. Biden and McCarthy counted on support from the political center, a rarity in divided Washington.

A compromise, the package restricts spending for the next two years, suspends the debt ceiling into January 2025 and changes some policies, including imposing new work requirements for older Americans receiving food aid and greenlighting an Appalachian natural gas line that many Democrats oppose. It bolsters funds for defense and veterans, and cuts new spending for Internal Revenue Service agents.

Congressman Chip Roy, a member of the Freedom Caucus helping to lead the opposition, said, “My beef is that you cut a deal that shouldn’t have been cut.”

For weeks negotiators labored late into the night to strike the deal with the White House, and for days Mr. McCarthy has worked to build support among skeptics. 

The speaker has faced a tough crowd. Cheered on by conservative senators and outside groups, the hard-right House Freedom Caucus lambasted the compromise as falling well short of the needed spending cuts, and they vowed to try to halt passage.

A much larger conservative faction, the Republican Study Committee, declined to take a position. Even rank-and-file centrist conservatives were unsure, leaving Mr. McCarthy searching for votes from his slim Republican majority.

Ominously, the conservatives warned of possibly trying to oust McCarthy over the compromise.

President Trump, for his part, held his fire: “It is what it is,” he said of the deal in an interview with Iowa radio host Simon Conway.

The House Democratic leader, Congressman Hakeem Jeffries, said it was up to McCarthy to turn out Republican votes in the 435-member chamber, where 218 votes are needed for approval.

As the tally faltered on an afternoon procedural vote, Mr. Jeffries stood silently and raised his green voting card, signaling that the Democrats would fill in the gap to ensure passage. 

They did, advancing the bill that hard-right Republicans, many from the Freedom Caucus, refused to back.

“Once again, House Democrats to the rescue to avoid a dangerous default,” said Mr. Jeffries.

“What does that say about this extreme MAGA Republican majority?” he said about the party, referring to Mr. Trump’s “Make America Great Again” political movement.

Then, on the final vote hours later, Democrats again ensured passage, leading the tally as 71 Republicans bucked their majority and voted against it.

The nonpartisan Congressional Budget Office said the spending restrictions in the package would reduce deficits by $1.5 trillion over the decade, a top goal for the Republicans trying to curb the debt load.

Liberal discontent, though, ran strong as nearly four dozen Democrats also broke away, decrying the new work requirements for older Americans, those aged 50-54, in the food aid program.

Some Democrats were also incensed that the White House negotiated into the deal changes to the landmark National Environmental Policy Act and approval of the controversial Mountain Valley Pipeline natural gas project. 

The energy development is important to Senator Manchin, but many others oppose it as unhelpful in fighting climate change.

In the Senate, Senators Schumer and McConnell are working for passage by week’s end.

Mr. Schumer warned there is “no room for error.”

Senators, who have remained largely on the sidelines during much of the negotiations, are insisting on amendments to reshape the package. 

Yet making any changes at this stage seemed unlikely with so little time to spare before Monday’s deadline.