He said the move would "lift the millstone of debt" from students.
Of the other contenders, Yvette Cooper also backs a graduate tax, Jeremy Corbyn wants to bring back maintenance grants and Liz Kendall would focus on early years education.
Mr Burnham also insisted his plan to "renationalise" the railways line-by-line could "finance itself".
Speaking on BBC Radio 4's  Today programme the shadow health secretary said money generated would "go back into supporting the railway system".
Under his plans, private companies could still bid to run franchises as they expire, but public bids would be encouraged and the branding of trains, as well as tickets and prices, would be controlled by a new National Rail organisation.
The pledges on rail and tuition fees feature in Mr Burnham's new manifesto, which also includes:
In its manifesto for the 2015 general election, Labour said it would cut tuition fees in England from Â£9,000 to Â£6,000 per year from autumn 2016.
It said the move would be paid for by reducing tax relief on pensions for those earning over Â£150,000.
Mr Burnham's team say the graduate tax will be considered in detail by a new Beveridge-style commission, which would look at who should cover the shortfall of the money needed by universities in the short-term.
At-a-glance profiles of the four contenders
Mr Burnham was also asked about the success of left-wing candidate Mr Corbyn's campaign, after the Islington North MP was nominated by more constituency Labour parties than any other.
"Jeremy has lifted the debate, and I think that's a good thing," he said.
The Leigh MP, who has previously floated the idea of a new tax to pay for the rising cost of social care, also said his promises would involve "difficult spending choices".
Andy Burnham is pledging to replace tuition fees with a "graduate tax" if he becomes Labour leader.