In an emotional speech, Starmer said he had listened to his governing Labour Party and realised that he was no longer the man who should lead it into a general election due in 2029.
After making his announcement on the steps of his Downing Street office and London residence, Starmer's move to stand down could have triggered a divisive leadership contest but several Labour MPs said they now expected more of a coronation.
Burnham, a 56-year-old career politician, quickly won the support of another potential leadership rival, former health minister Wes Streeting, with one Labour MP saying it was more likely the former Greater Manchester mayor would be installed as leader.
He entered parliament to cheers from Labour MPs, and the odd heckle from the opposition, including one person who shouted out: "He's not the Messiah."
He then pledged allegiance to the monarch to take his place as an MP, a necessary step to be able to become the new leader of Labour and the United Kingdom.
The Labour government is the latest to fall foul of voter anger over politicians' failure to deliver on their promises of change, 10 years after the vote to leave the European Union.
Starmer said he would ask the Labour Party's organising committee to set out a timeline for a leadership contest to find his replacement.
Nominations would open on July 9, close by mid-July and if there is a contest, a new leader will be in place by September.
A coronation could mean a new leader would enter office by mid-July.
Burnham said on Monday there were many steps ahead when asked if he planned to call a general election.
When asked by a BBC journalist if he would call an election as Labour leader, Burnham said: "You're jumping several hurdles ahead there."
Burnham is the only MP to date to have confirmed he will enter the leadership contest.
Streeting said on Monday Burnham had not offered him the finance minister role or any other job in a future cabinet.
"He has not offered me any jobs," Streeting told the BBC.
"That's not what our conversations have been about."
Starmer announced on Monday morning that he would resign after spending the weekend with his wife at his country residence considering his future.
"The question my party is asking now is whether I am best placed to lead us into the next general election," Starmer said in his address.
"I have heard the answer from my parliamentary party to that question and I accept that answer with good grace," he said.
After describing the achievements his government had secured in his two years of power, a man who was often criticised for being robotic became visibly emotional, his voice cracking when he thanked his family for their support.
"When I leave the biggest job in the country I will spend more time on the most important job, being the best husband I can to my fantastic wife Vic who has been a rock by my side through good times and bad, and being the best dad I can to my beautiful children who are my pride and my joy."
There was some sadness in the Labour ranks, with industry minister Chris McDonald saying his speech underlined the fact "he's a really decent man".
However, others said he had been treated the way he treated others as prime minister, being "royally done over".
Despite hoping for a smooth handover, the change is not without risk.
Burnham has yet to flesh out a full policy agenda and opposition party Reform UK leader Nigel Farage called for an election.
"I've had enough of waiting around. Britain needs change - real change, not another washed-up has-been shoved into place by the uniparty," Farage said.
with AP
