The new mayor insisted that his absence was “not in the slightest” intended as a snub, saying he had told Mr Corbyn’s office earlier in the week that he had prior engagements which would prevent him from appearing alongside the Labour leader.
He denied Labour was split, insisting the party was “fighting as one” for the June 8 General Election, and promised he would be out on the campaign trail in the coming days.
He said that Labour should learn from the “grassroots” campaigning which helped him to a victory which was one of the party’s few bright moments on a day of election gloom.
There was widespread surprise that Mr Burnham was not present when Mr Corbyn addressed supporters on the steps of Manchester Convention Centre on Friday, hours after the election result was announced at the venue.
Mr Corbyn said he had spoken to the new mayor who was “busy working for Manchester already”.
But photographs circulating on social media showed Mr Burnham apparently celebrating the result in a restaurant with supporters.
Meanwhile, the Labour candidate for Manchester Central, Lucy Powell, said she was not told in advance of Mr Corbyn’s visit, tweeting: “I didn’t get invited and it’s my constituency”.
Ms Powell later said that Mr Corbyn’s office had apologised for what was “a simple mistake”.
Mr Burnham told the BBC: “I had made it clear earlier in the week that I wouldn’t be able to be at the rally at seven o’clock, because I had a lot of commitments, including family commitments.
“Jeremy came - fair enough, because people wanted to enjoy the moment.”
The former Leigh MP and two-time Labour leadership contender, who is standing down from Parliament, said: “For me, this kind of row - Was he at something? Was he not at something? What does this mean? - is the politics I’m leaving behind, that kind of trivia. I’m fed up of it, I don’t think the public care.”
Mr Burnham said he found it “humbling” to win the first election to the Greater Manchester metro mayoralty with 63 per cent of the vote.
“It was a very mixed picture for the party yesterday,” he said.
“I would say the party needs to do what we are doing here, which is to go back to grassroots and build from there.
“The support I received was truly very humbling and hopefully is a reflection of the work we have done here over many years, building that connection with people at a very local level.”
On his first full day in office, Mr Burnham announced the appointment of Manchester City Council leader Sir Richard Leese as his deputy mayor for business and the economy and former immigration minister Baroness (Beverley) Hughes as deputy mayor for policing.
Bury council leader Rishi Shori will be portfolio holder for young people and social cohesion.