The success of the Islington North MP prompted another day of drama in the leadership campaign, as voting got underway.
Former foreign secretary David Miliband warned Mr Corbyn would take the party backwards as he endorsed Liz Kendall for the top job to be unveiled on September 12.
But Ms Kendall and the other candidate for leader, Yvette Cooper, were both forced to rule out standing aside to enhance Mr Burnham’s hopes of thwarting the left-wing challenger.
Both Mr Burnham and Mr Corbyn offered to “involve” each other in their teams if they succeeded Ed Miliband.
Mr Burnham said there is a “good deal of common ground” between him and his rival on major policy areas, including transport and education.
As he reached out to the Islington North MP’s supporters, the shadow health secretary said Mr Corbyn had brought “real energy to this race” and he would “involve” his rival in his team.
But Mr Burnham warned there is a “real risk of division” following the increasingly bitter leadership contest and insisted he was the one candidate who could unite the party.
He said it was crucial for Labour to have a “credible” plan for public finances and criticised the Islington MP’s opaque stance on Britain’s future in the European Union as well as his plans for renationalisation of utilities and “printing money” to pay for infrastructure.
Speaking at the People’s History Museum in Manchester yesterday he said: “My worry is that policies like these would leave us open to losing the argument on the economy on day one.”
He added: “I won’t let our party repeat the history of the early 1980s when we were more interested in fighting each other and we left the pitch clear for Margaret Thatcher to bulldoze her way through Labour communities up and down the country.
“I won’t let those Bullingdon Boys - the heirs to Thatcher - do the same to people in 2015 as she did in 1985.
“That was the year I joined Labour here in the North West. I have given my life to it ever since.
“I am now fighting to win this contest with everything I’ve got because the very future of our party hangs in the balance.”
After the speech, Mr Corbyn said: “We welcome Andy’s inclusive tone towards our campaign and the view is mutual - if we win we would involve Andy in our team if he was willing. From day one, whoever wins must pull the party together.”
It came as the latest of a string of interventions from the party’s big beasts saw Mr Miliband endorse Liz Kendall to take the top job, while his second preference vote would go to Yvette Cooper.
“The Corbyn programme looks backwards,” he added in a newspaper article.
“The pledges of nationalisation, 7p in the pound increases in national insurance for those earning more than £50,000, and equivocation about Britain’s place in the EU are the same ideas that I learned were wrong when I joined the Labour party in 1981.”
Mr Miliband dismissed the radical left-winger’s “demand that Labour become an anti-austerity movement on the Greek model”.
Ms Cooper and Ms Kendall have denied reports that Lord Mandelson approached them in a bid to suspend the election by persuading the pair and Mr Burnham to drop out en masse.
While Ms Cooper said she had not been approached directly by the grandee and said she was unaware if her campaign team had been contacted, Ms Kendall insisted that neither she nor her office had spoken to the peer.
Asked about the claims made by the Telegraph, Ms Kendall told BBC Radio 4’s World At One: “No, neither me nor anybody in my team (have spoken to Lord Mandelson). I have no idea where that came from.”
Ms Kendall also rubbished suggestions that she has been asked to drop out of the race to boost Mr Burnham and Ms Cooper’s chances of winning.
She said: “I haven’t made that proposal and none of the other candidates have made that to me.
“I have said that I want people to vote for me with their first preference but they should use their second or third preferences for Yvette or Andy or Andy or Yvette.
“I’m not going to be dropping out in this contest, I can’t stop making the case.”