The MP for Makerfield cemented his status as the sole leadership candidate after being backed by 379 Labour MPs, as well as all 11 trade unions affiliated to the party, earlier this week.
He will take over from Sir Keir as prime minister on Monday, following a reception with King Charles III.
Asked by reporters about who will be in his cabinet, Burnham said: “I am finalising those decisions, and I will come to conclusions very shortly, and then I will announce those on Monday.”
In his first speech as Labour leader at the TUC headquarters in central London, he said his appointments for cabinet would reflect “all parts of our party” and “all communities”.
He said his government would give “hope back” to working-class communities who the political class had “turned its back on”, reiterating his plan to devolve power away from Westminster to the regions.
Devolution has been one of Burnham’s central messages since he announced his bid to replace Sir Keir.
He said his government would deliver the biggest change in decades, rejected the “neoliberal” policies of the past 40 years, including when he was in government, and vowed to reindustrialise the country.
And in a message to Labour MPs, he said he would work relentlessly to build a culture of “one Labour team”.
“We won’t beat Britain’s new right if we are consumed by infighting and pulling in different directions,” he said.
He added that his approach to governing would be characterised by “problem-solving rather than point-scoring”.
He added that he knows what he believes and what he wants to do in government, telling the audience: “I have a plan.”