Breaking with President Trump’s drive to overturn his election loss, Senator Mitch McConnell of Kentucky on Tuesday congratulated President-elect Joseph R. Biden Jr. on his victory and began a campaign to keep fellow Republicans from joining a doomed last-ditch effort to reverse the outcome in Congress.

Although Mr. McConnell waited until weeks after Mr. Biden was declared the winner to recognize the outcome, his actions were a clear bid by the majority leader, who is the most powerful Republican in Congress, to put an end to his party’s attempts to sow doubt about the election.

He was also trying to stave off a messy partisan spectacle on the floor of the House that could divide Republicans at the start of the new Congress, forcing them to choose between showing loyalty to Mr. Trump and protecting the sanctity of the electoral process.

“Many of us hoped that the presidential election would yield a different result, but our system of government has processes to determine who will be sworn in on Jan. 20,” Mr. McConnell said in a speech on the Senate floor. “The Electoral College has spoken. So today, I want to congratulate President-elect Joe Biden.”

A short time later, on a private call with Senate Republicans, Mr. McConnell and his top deputies pleaded with their colleagues not to join members of the House in objecting to the election results on Jan. 6, when Congress meets to ratify the Electoral College’s decision, according to three people familiar with the conversation, who described it on the condition of anonymity.

A small group of House members, led by Representative Mo Brooks of Alabama, plans to use a constitutional process to object to the inclusion of five key battleground states that day. There is almost no chance they will succeed. But if they could persuade at least one senator to join them, they could force a vote on the matter, transforming a typically perfunctory session into a bitter last stand for Mr. Trump.

So far, no senator has committed to joining them. In seeking to prevent anyone from doing so, Mr. McConnell argued that a challenge would force senators to go on the record either defying Mr. Trump or rejecting the will of the voters, potentially harming those running for election in 2022.

He dispatched his top deputy, Senator John Thune of South Dakota, to lobby lawmakers one by one.

The remarks were a decisive shift for Mr. McConnell. They came only after members of his leadership team in the Senate — and even the chamber’s chaplain — began softening the ground by congratulating Mr. Biden on Monday evening and Tuesday morning.

Though he never repeated them, Mr. McConnell had allowed Mr. Trump’s false allegations of election fraud and fantastical claims that he had been the true winner to circulate unchecked for more than a month, defending the president’s right to challenge the election outcome in court. Allies insisted privately that he would ultimately honor the results, but did not want to stoke a year-end conflict that could hurt the party’s chances in two Georgia Senate runoffs and imperil must-pass legislation.

That calculus changed late Monday, after electors across the country cast their ballots for Mr. Biden, cementing his 306 to 232 Electoral College victory. By Tuesday morning, Mr. McConnell and his leadership team were openly acknowledging the results and creating the political space for other Republicans to begin belatedly recognizing Mr. Biden as the winner.

The Senate leader also spoke by phone with Mr. Biden, apparently for the first time since his former Senate colleague won the presidency more than five weeks ago.

“I called to thank him for the congratulations, told him although we disagree on a lot of things, there’s things we can work together on,” Mr. Biden told reporters, adding that it was a “good conversation.”

In normal times, such a call would have drawn little notice. But Mr. Trump’s push to deny his loss has created a charged political moment for Republicans, spotlighting rifts within the party and placing Mr. McConnell in a particularly dicey position.

Polls suggest a clear majority of Republicans believe Mr. Trump’s fabrication that the election was fraudulent, and they are likely to follow the president’s words, not those of Mr. McConnell. Meanwhile, many of the president’s allies in the House continue to support his challenges to the results, with more than 60 percent of them signing on last week to a legal brief endorsing the failed effort by Texas to overturn results in key battleground states. The House’s top leaders were mostly silent on the question on Tuesday, and their aides did not respond to questions about Mr. Biden’s victory.

Mr. Trump himself showed no signs of backing down, repeating his false allegations on Twitter just after Mr. McConnell spoke: “tremendous evidence pouring in on voter fraud.” Mr. Trump also shared a news article about Mr. Brooks’s efforts, raising the possibility that he could begin pressuring members of the party to join in, stoking an even bigger fight in the weeks ahead.

Moving to head off potential backlash, Mr. McConnell told reporters pressing him to rebuke Mr. Trump’s rhetoric that he did not have “any advice” for Mr. Trump. Earlier in the day before congratulating Mr. Biden, he had used his speech on the Senate floor to lavish praise on the president’s record on foreign and domestic policy.

The blowback was immediate from the party’s outspoken right flank anyway and foreshadowed the return to an old dynamic briefly abated during the Trump years in which Mr. McConnell was a favorite punching bag for conservatives. Mark Levin, the talk radio host and strident supporter of Mr. Tump, declared that Mr. McConnell had been “AWOL” from “challenging the lawless acts of the Biden campaign and Democrats.”

“Trump helped you secure your seat, as he did so many Senate and House seats, and you couldn’t even wait until January 6th,” Mr. Levin wrote on Twitter. “You’ve been the GOP ‘leader’ in the Senate for far too long. It’s time for some fresh thinking and new blood.”

Nor did Mr. McConnell earn much love from the few voices of Republican dissent that have raised alarms in recent days that Mr. Trump’s defiance of democratic norms — and the acquiescence of much of his party — would do lasting damage both to the G.O.P. and to the country.

One of them, Representative Paul Mitchell of Michigan, who is retiring, went as far as to quit the party on Monday in protest. Another, Senator Mitt Romney of Utah, the party’s 2012 presidential nominee, said on Tuesday that merely recognizing Mr. Biden’s victory was not enough for his party.

“How many Republicans will say that what the president is saying is simply wrong and dangerous?” Mr. Romney said on CNN. “We need to have people who are strong Trump supporters say that as well, or you are going to continue to have this country divided, which is pretty dangerous.”

Representative Tom Cole of Oklahoma, a senior Republican committee leader and former head of the party’s campaign arm, argued that fears like Mr. Romney’s were somewhat overwrought, but reflected a general loss of trust by many Americans in the electoral process.

“We need to accept our institutions. They worked in 2016,” Mr. Cole said in an interview. “They worked again in 2020.”

Elected officials, he said, need to “be honest with your voters.”

“You have to recognize when you are not successful, and you move on and accept the election results,” he said. “The American people, I hope, will do that.”

In the Senate, at least, that view appeared to be gaining currency.

In a statement, Senator Mike Lee, Republican of Utah and a defender of the president, said that “absent new information that could give rise to a judicial or legislative determination altering the impact of today’s Electoral College votes, Joe Biden will become president of the United States.” An aide said he had no plans to join Mr. Brooks in challenging the results.

Another leading contender for that task, Senator Ron Johnson, Republican of Wisconsin, seemingly threw cold water on the idea as well. Though Mr. Johnson plans to convene a hearing on Wednesday to give Mr. Trump’s specious arguments of voting fraud an airing in Congress, he told the Milwaukee Journal Sentinel that he thought the outcome was legitimate and he did not plan to object to the Electoral College results.

Still, other possible contenders remained. One was Tommy Tuberville, the newly elected Alabama Republican. Another possible candidate, those watching the process said, was Senator Kelly Loeffler of Georgia, one of the two Republicans competing in January runoffs that will determine which party controls the Senate next year. Those races will play out the day before the joint session to ratify the presidential election results convenes in Washington.

Ms. Loeffler’s office did not respond to a question about Mr. Biden on Tuesday, but on Twitter, she suggested she was not ready to accept the result.

“I will never stop fighting for @realDonaldTrump because he has never stopped fighting for us!” she wrote.

Luke Broadwater and Emily Cochrane contributed reporting.