Burnham, the Greater Manchester mayor nicknamed the "King of the North", won the contest on Friday in Makerfield in northwest England with 24,927 of the votes, while the candidate for Nigel Farage's populist Reform UK party came second with 15,696 votes.
His victory means he will now be able to trigger or at least take part in a contest to replace Starmer, struggling with some of the worst popularity ratings of any leader.
But the key question is when and how he will do it.
"When this borough went to the polls in May, it made a loud cry for change. In this campaign we have begun to answer that," Burnham said in his victory speech.
"But I do say to my own party: this is a final chance to change. This is what people said directly to me on the hundreds of doorsteps that I stood on. We must hear it, we must act upon it and we must get it right. There will be no second chance.
Burnham, a career politician who has expressed support for the nationalisation of key public services and criticised what he called four decades of failed neoliberal economics, has said he would seek to replace Starmer and change politics.
"But there is a chance now from this result tonight to build a new politics based on unity and hope. Turning away from the path that takes us to a divided, dark politics of the kind we see in the United States," Burnham said.
"We must now take this path and put this country back on the right path and bring people back together and get things working properly again."
Polls show Burnham, 56, is Labour's most popular politician who would win a months-long leadership contest decided by party members, while some Labour lawmakers hope Starmer could be persuaded to hand over power to avoid a damaging contest.
Two years after winning a landslide national election, Starmer, 63, is one of the least popular British prime ministers since polling records began after scandals, policy U-turns, and accusations of indecision have derailed the delivery of the change he once promised.
About a quarter of Starmer's lawmakers have called for him to quit since his party suffered heavy losses in local elections in May, while senior colleagues, including the defence and health minister, resigned in recent weeks over his leadership.
A defiant Starmer this week said he would stand in any leadership contest and issued a warning to his party about the potential "chaos" of a potentially bitter and divisive leadership election.
Another of Starmer's main rivals, former health minister Wes Streeting, said this week he would force a contest soon unless the prime minister announced when he would stand down.
Under Labour rules, it would take 20 per cent of the parliamentary party, or 81 lawmakers, to announce they were backing a single candidate to trigger a leadership challenge.
