The shadow home secretary used his victory speech to send a message to the Prime Minister, saying: "You and your party have made promises to the people here in the North of England and I will not let you walk away from them."
Mr Burnham pledged he would be a mayor to hold the Government's feet to the fire over investment in the North, which he said had been grotesquely unfair.
He said: "You promised us a powerhouse, we now demand it."
Mrs May's unpicking of George Osborne's legacy had sparked speculation she had gone cold on the idea of a Northern Powerhouse.
Her decision to stall on the £18bn Hinkley nuclear deal, angering Chinese investors, raised concerns there could be ramifications for Chinese investment in northern infrastructure projects.
The MP for Leigh said he would end London-centric politics, close the North-South divide and stop the "voice of Scotland" drowning out the "voice of the North".
And in promising to be a "grassroots mayor" he said he would establish a successful comprehensive schooling system marking his fierce opposition to the idea of bringing back grammar schools.
However, Mr Burnham said he would be standing down as MP, and therefore from the shadow cabinet, at the "earliest opportunity" if he became mayor.
Manchester is a Labour stronghold so it is likely that Mr Burnham will triumph in next year's vote, finding himself in charge of a £6bn health and social care budget and a £900m, 30-year investment fund.
Mr Burnham beat interim mayor Tony Lloyd and former Labour government minister Ivan Lewis to become the Labour candidate.
It has been suggested that Mr Burnham's departure from the Westminster scene could be part of a "brain drain" of top Labour figures who, under Jeremy Corbyn's leadership, no longer feel they can influence the direction of the party.
Former shadow minister Luciana Berger is in the running to be Labour candidate for Liverpool Mayor.