WASHINGTON — When 14 Senate Republicans joined forces with Democrats on Thursday to pave the way for Congress to avert a first-ever federal default, it reflected the crucial role of the pragmatic wing of the G.O.P. in a divided government. But it also showed how narrow that wing has become, and how willing the majority of Republicans were to use potential fiscal catastrophe as an opening to pummel President Biden and his party.

In the not-too-distant past, the debt limit escape plan concocted by Senator Mitch McConnell, Republican of Kentucky and the minority leader, and Senator Chuck Schumer, the New York Democrat and majority leader, would have been applauded by Republicans grateful for Mr. McConnell’s dexterity in getting them off the political hook on a tough issue.

But in today’s Republican Party, it prompted considerable blowback from Mr. McConnell’s usually unquestioning colleagues. That is a mark of how far the party has strayed from what it once was, and reflects a growing divide in the Republican ranks, one likely to deepen next year if Republicans win control of the House and Senate through the election of new members who emerge from primaries that reward the furthest-right — and most uncompromising — candidates.

Under the agreement struck between the two leaders, Congress approved an entirely new law — good for one-time use only — to set a simple Senate majority threshold for an increase in federal borrowing authority, clearing the way for Democrats to raise the debt ceiling on their own next week without threat of a Republican filibuster. The legislation advanced Thursday on a 64-to-36 vote, with Mr. McConnell and 13 other Republicans joining all 50 Democrats in support of taking it up while 36 Republicans objected. The measure was then approved and sent to Mr. Biden.

It was a classic McConnell gambit to find a way to allow the debt ceiling to be raised while simultaneously allowing all Republicans to vote against it.

But most Republicans in the Senate and virtually all of them in the House, fearing primaries from the right and harsh condemnation from Mar-a-Lago, wanted nothing to do with the legislative sleight of hand. They denounced any move to allow the debt ceiling to be raised as complicity with Democrats’ multitrillion-dollar economic agenda — even though the increase is needed mainly to cover spending on existing programs, some of which Republicans supported.

Republicans howled that Mr. McConnell should have stuck to his guns and forced Democrats to act entirely on their own, claiming that he had established a dangerous precedent by giving the House a role in circumventing Senate filibuster rules.

“What we’ve done here is allowed the House to change the Senate rules in a fashion where if you can get 10 Republicans, all of us are dealt out,” said Senator Lindsey Graham, Republican of South Carolina. “That is not a good idea one time, 10 times or a hundred times — by either party.”

Other Republicans said Mr. McConnell had put his own members in a political bind by combining the debt limit proposal with a measure to restore Medicare cuts — included as a sweetener to generate support, forcing anyone voting against the debt ceiling bill to also go on record as favoring cuts to Medicare, potentially upsetting older voters.

“The disagreement that reasonable people are having over the debt limit has been conflated in a cynical attempt to fool the American people,” said Senator John Kennedy, Republican of Louisiana, who said members of his party had been given a “choice between voting for a heart attack or cancer.”

But with the debt limit so inflammatory among hard-right conservatives as a proxy for Democratic spending plans, most Republicans chose to reject the legislation and risk being attacked on Medicare spending, figuring they could explain their way out.

The dispute highlighted how profoundly today’s Republican Party is motivated by concern about being seen as accommodating Mr. Biden or congressional Democrats, suggesting difficulties ahead should the party take control of the Senate and it falls to them to increase the debt limit and perform the other basics of governing.

Complicating matters is the fact that some of the seasoned Republicans who helped Mr. McConnell in this case — Senators Roy Blunt of Missouri, Rob Portman of Ohio and Richard Burr of North Carolina — are leaving Congress at the end of next year, shrinking the ranks of those willing to risk political blowback in the interest of keeping the government from falling into chaos.

If Mr. McConnell is struggling to corral the current members of his Republican conference after 2020 elections that added more hard-right Republicans, it is likely only going to become more difficult when some of his most dependable allies are gone, potentially replaced by archconservatives fueled by antipathy to government and fealty to Donald J. Trump.

The ongoing transformation of the Republican Party was underscored even further on Thursday by the fact that lawmakers were conducting memorial services in the Capitol Rotunda for Bob Dole, the former Senate Republican leader, presidential nominee and deal maker if there ever was one.

Despite the bad optics of the Republican disagreement, Mr. McConnell’s supporters — and even some of those unhappy with his strategy — said he had made the smart move in finding a way to give Republicans an off-ramp.

“He is looking at the big picture, and he is looking at what is best for the country in terms of how we get past a crisis,” said Senator Mike Rounds, Republican of South Dakota, who opposed the debt limit bill.

The Republican leader’s chief miscalculation, several said, was declaring in October that neither he nor any other Republican would help raise the debt ceiling again after allowing a temporary increase to move forward — a stance from which he was forced to back down.

But allies said he had little choice. Given the current struggles of the Biden administration, Republicans see themselves in a strong position to win back control of Congress next year, and Mr. McConnell and his top lieutenants were reluctant to give Democrats any ammunition to turn the tide.

They saw failure to raise the debt ceiling and the ensuing economic fallout as a potential political liability that Republicans had to avoid, even if they were uncomfortable doing so.

“It makes no sense to change the subject,” said Senator John Cornyn, Republican of Texas and one of those who voted to advance the legislation. “When your adversary is imploding, why interrupt him?”

Republicans also noted that the agreement preserved the Senate filibuster, a chief objective of Mr. McConnell. He feared that continuing the Republican blockade against raising the debt limit would force the hand of the few remaining Democratic holdouts against weakening the filibuster, an outcome he wanted to avoid at all costs. While the legislation created a one-time exception from the requirement to secure 60 votes to overcome a filibuster, it did so only with Republican consent and cooperation.

Lastly, the agreement will require Democrats to set a specific amount for the increase rather than something more amorphous, providing what Mr. McConnell and other Republicans saw as a good cudgel for the midterm elections as they hammer Democrats on spending and inflation.

In the end, Mr. McConnell hopes that the machinations and infighting will be forgotten, and that voters will remember only that Democrats increased the federal debt ceiling themselves by hundreds of millions — possibly trillions — of dollars, sparking a backlash that restores Republicans to power.

Then Mr. McConnell will have to figure out how to govern with colleagues who seem increasingly uninterested in doing so.