Congress approved a massive increase in the federal debt Wednesday, but only after Republican leaders linked arms and jumped together, helping Democrats pass a plan that will let President Obama borrow as much money as needed to cover federal obligations over the next 13 months.

The vote was about as dramatic as they get. Senate Minority Leader Mitch McConnell, Kentucky Republican, held out for 40 minutes before casting the key vote to break his fellow Republicans’ filibuster. His top lieutenants then followed suit, clearing the way for a final vote to approve the debt ceiling increase.

The bill passed the House on Tuesday. It now goes to Mr. Obama, who is expected to sign it.

Democrats warned of disastrous consequences if the limit hadn’t been raised, saying the government would default on its obligations. Analysts said a failure would have been devastating for the stock market, which was closely watching the vote.

Senate Majority Leader Harry Reid, Nevada Democrat, said he hoped the fact that Republicans approved a debt increase would be a breakthrough for other issues.

“Congress should be striding from accomplishment to accomplishment, not staggering from crisis to crisis,” he said. “If we spent more time working together and less time running out the clock on procedural hurdles and Republican filibusters, we might actually get things done around here.”

But Republicans who opposed the debt limit increase said it marked a low point for Congress and for their party in particular, which could have blocked the increase if the party’s lawmakers held together.

“Let’s be clear about the motive behind this vote — there are too many members of Congress who think they can fool people and they will forget about it the next week,” said Sen. Ted Cruz, the Texas Republican who led the attempted filibuster. “But sometimes, come November, the people remember.”

Indeed, in the minutes after the vote, conservative groups blasted out emails criticizing Republicans who enabled the debt increase, Mr. McConnell in particular.

The Kentucky Republican faces a challenge in a party primary this year from Matt Bevin, who said the vote showed the failure of Mr. McConnell’s leadership.

Three years ago, after Republicans won major spending cuts attached to the 2011 debt increase, Mr. McConnell said they set a strict standard.

But the latest debt increase comes with no strings attached — the first time that has happened since Republicans won control of the House in the 2010 elections.

The increase lasts until the middle of March 2015, and there is no set limit. Instead, the debt will rise by as much as is needed to cover obligations. Whatever the number is as of March 15, 2015, will become the new legal limit.

As of Monday, the federal debt was $17.259 trillion, where it has been frozen since last month when the previous limit was reached. Federal debt was $10.629 trillion when Mr. Obama took office.

The Treasury Department has said it would run out of maneuvering room on the debt by the end of this month.

Despite the drama Wednesday, the increase in the debt limit was much easier than previous votes — chiefly because Republicans surrendered early in hopes of attaching conditions.

The outcome was assured — the only question was which Republican would have to cast the politically perilous vote. During the hourlong process, Republicans gathered in small groups on the Senate floor to discuss the matter.

Senators said four Republicans were willing to join all 55 Democrats, but that left them one vote shy of the 60 needed to break the filibuster.

Ultimately, Mr. McConnell came to the well of the chamber and cast his vote as the crucial 60th supporter.

Sen. John Cornyn of Texas, the second-ranking Republican, then switched to join Mr. McConnell in opposing the filibuster. He and other allies then went to try to persuade other Republicans to join them in a show of solidarity.

“All right, let’s go — come on down,” Sen. Bob Corker, a Tennessee Republican who voted “yes” to break the filibuster, told his colleagues.

But few followed. The final tally was 67-31. Minutes after breaking the filibuster, the debt increase was approved on a party-line 55-43 vote.

Mr. Corker criticized Mr. Cruz for forcing such a politically dangerous vote.

“Cloture is to end debate. There was no alternative offered. We knew the House could only pass a clean [debt increase], so why do you debate it any longer? It’s either a yes-or-no vote,” he said.

House Republican leaders passed the debt increase on a 221-201 vote that relied almost entirely on Democrats. Just 28 Republicans voted for it.

The GOP was fractured on the issue, and many were spooked after being blamed for the partial federal government shutdown in October. Democrats were united against any concessions, convinced that Republicans would be blamed for a breach of the debt ceiling.

Republicans said they agreed to the increase to avoid detracting from their message against Obamacare, which they believe will help them regain control of the Senate in November’s elections.

“Our focus really continues to be on Obamacare and other issues,” said Sen. John McCain, an Arizona Republican who initially supported the filibuster but switched his vote after Mr. McConnell voted. Mr. McCain called Mr. McConnell’s vote “a very courageous act,” given his tough re-election race.

Mr. McConnell didn’t issue any statement about his vote.

Afterward, the Senate wrapped up business and went on recess for 11 days.

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