In brief remarks, a beaming Mr. Burnham said people had “voted for change, they have voted for more power for the north, they have voted for hope.” It is a message he intends to take to his bid to become prime minister.

“I do say to my own party, this is a final chance to change,” Mr. Burnham said. “We must hear it, we must act upon it, and we must get it right. There will be no second chance. But it is a chance now, from this result tonight, to build a new politics.”

Mr. Starmer congratulated Mr. Burnham on his victory on Friday morning but repeated his vow to resist any challenge to his leadership. Speaking to the BBC, Mr. Starmer said an intraparty contest would plunge Britain “into chaos.”

“If there is a contest, then yes I will run, I will stand, and I’ve said repeatedly I’m not going to walk away from that,” he added.

Reform U.K., which is led by Nigel Farage, an ally of President Trump, failed in its attempt to thwart Mr. Burnham, in spite of its success in a set of local elections last month. Mr. Kenyon did better than his party did in the general election two years ago but came in second with 15,696 votes, or about 34 percent of the ballots cast.

Election officials announced the results of the special election in Makerfield, an area made up of former coal mining villages and market towns, on Friday morning.

The win will galvanize supporters of Mr. Burnham, who have argued that he offers Labour its best chance of challenging Reform. In his remarks, Mr. Burnham said his victory was a chance for the country to turn “away from the path that takes us to a divided, dark politics of the kind we see in the United States.”

Mr. Kenyon was partly undercut by the presence of a candidate representing Restore Britain, a far-right party that argues that Reform is not extreme enough. The Restore candidate won 3,111 votes, or almost 7 percent of the ballots.

But added together, the vote for the two right-wing parties would have still fallen far short of the number needed to defeat Mr. Burnham and win the race.

Mr. Burnham may now begin the process of trying to oust Mr. Starmer, who has become one of the least popular prime ministers in modern British history.

When Mr. Burnham might challenge the prime minister remains unclear. Several Labour lawmakers have publicly said Mr. Starmer should step aside, for the good of the party and the country, if Mr. Burnham challenges him.