In front of dozens of supporters outside his new office and residence at 10 Downing Street, Burnham said politicians had failed to bring the stability needed to raise living standards, something he said he would change with immediate policies and a longer-term plan to be drawn up later this year.
"I know people at home are fed up with politics. I hear you, and I want to be honest with you, we have not been good enough and we need to be better," he said, speaking without notes and dispensing with the lectern that prime ministers generally use.
"We will make this moment a circuit-breaker for Britain, bringing forward the biggest changes in the last 40 years. A new political model and a new economic model."
Just over an hour after his predecessor and Labour colleague Keir Starmer made his farewell speech to say "My work is done," the 56-year-old former Greater Manchester mayor entered Downing Street, where he said his team would tackle his first priority: to end rough sleeping.
But the major challenges lie ahead.
Burnham must first unveil his cabinet - already the subject of much debate in the Labour party - and then tackle a long list of problems ranging from anaemic economic growth to a cost-of-living crisis and underperforming utility firms.
Starmer, who was ousted by his own MPs about a month ago, said his work as prime minister was "the privilege of my life" and that he wished Burnham every success.
"He has my full support," Starmer said.
Burnham did not refer to Starmer in his speech, instead focusing on how he will rip up much of what went before.
But he acknowledged that some of his program would take time.
"Later this year, I will bring forward a new Plan for Britain, a 10-year plan, laying out a plan from where we are now to where I believe we all want Britain to be, wherever we're coming from, whatever party we support," he said.
But he said he would also give people "some breathing space now" by setting out measures on Tuesday to tackle the cost of living, and how he will pay for them.
Beyond tackling homelessness, Burnham said he would build more public housing to help bring down welfare spending, which would help fund higher defence investment.
On Friday, Burnham described his election to become Labour leader as "the most significant change moment in our politics for 40 years".
He promised to radically change the political system to help a country hungry for change become one where "life is more affordable and all people and places are lifted from where they are now".
Labour MPs view him as one of the few politicians capable of tackling the threat from veteran Brexit campaigner Nigel Farage's populist Reform UK - something they doubted the unpopular Starmer could do.
Burnham's first challenge will be the choice of his cabinet team and especially his finance minister.
Friction in this axis of government has brought down previous administrations.
An early frontrunner for the position, energy security and net zero minister Ed Miliband, has been the object of some hostile briefing and interior minister Shabana Mahmood now appears favourite for the job.
On Sunday, Burnham scrapped plans for all employees to be required to hold a digital identity document, a scheme designed to tackle illegal migration but deemed a "fiasco" by a cross-party committee of MPs.
More attention will be paid to decisions on taxation and spending, oil and gas, and underperforming utility companies, where Burnham wants stronger public control.
