His comments came in the first televised debate, in the marginal constituency of Nuneaton, where Labour lost to the Tories at the last election.
Mr Burnham faced off with his three rivals – shadow home secretary Yvette Cooper, the standard bearer for the Blairite right of the party Liz Kendall and veteran left winger Jeremy Corbyn.
During the debate, a member of the public, Monique Morris, asked: “What qualities do you share with Nicola Sturgeon that could make you as successful as a party leader?”
The question referred to the huge profile now enjoyed by Ms Sturgeon after leading her party to win 56 of Scotland’s 59 seats followed by a highly successful trip to the USA.
The only direct answer came from Mr Burnham, who praised Ms Sturgeon as “a plain talker” with strong values. He said: “She speaks in a language people understand. I think she came into politics because she saw an unfair country and that’s what brought me into politics.”
Earlier, Mr Burnham had led attacks on the way Prime Minister David Cameron had used the election to stoke up English nationalism.
He warned the country is a “divided Britain” after Mr Cameron’s English nationalist election campaign. He accused Mr Cameron of “stoking up separation” and promised that he would “bring the country back together”.
Ms Cooper used the question to appeal for Labour to have its first woman UK leader more than 40 years after the Tories elected Margaret Thatcher.
She said: “I have a great respect for Nicola Sturgeon. I think as a party which has campaigned for women’s rights for a century it would be great to smash that last glass ceiling and have a woman leader.” Ms Kendall refused to refer to Ms Sturgeon, but said: “Leaders are tough, they say things that are difficult, not easy – the best leaders set out a compelling vision of a better life.”
Meanwhile, Mr Corbyn said: “I have shared platforms with Nicola Sturgeon in opposing Trident and the Iraq war.”
He added: “Our fundamental mistake was the Better Together campaign joining with the Tories instead of putting forward our own separate Labour views.”
But despite the SNP running a more left-wing campaign than Labour in the last election, he said: “We need to be more radical left than the SNP.”
Mr Burnham appealed for the party to move away from the Blair years. The shadow health secretary said the former prime minister “did a lot of things right” and had reached out to voters who wanted to “get on in life”.
He promised to end the party being “run by an elite” and “get it out of the Westminster bubble”.
However, Ms Kendall, the candidate most closely identified with the Blairite strand, insisted she was “not Blairite, Brownite, Old Labour, New Labour”.
She promised to be “the fresh start leader” warning that Mr Burnham and Ms Cooper “carried baggage” from the Blair/ Brown years.