Insisting that a British exit would put jobs and the economy at risk, the shadow health secretary said it was time to "nail our colours firmly to the mast" and pledged to be "proudly pro-European" if he takes the top job.
The warning comes after Corbyn suggested that Labour could hold a special conference before deciding its response when Prime Minister David Cameron puts forward his package of reforms on Britain's relationship with Brussels ahead of the in/out referendum.
At a campaigning event in Swansea, Burnham said: "In just over a year, Britain faces a momentous decision that will affect every single person in our country; a decision that will define our politics and our country for the rest of this century.
"The stakes couldn't be higher and this is no time for equivocation. To flirt with exit from the EU is to put people's jobs, communities and future prosperity at risk. This is no time to cut ourselves off from our largest market. It is time to nail our colours firmly to the mast.
"Any Labour Party I lead will be proudly pro-European and will instantly set about the task of securing our membership of the EU. I will build a grassroots movement in workplaces across the country to give voice to the pro-European case and counter the double-speak of Tory Eurosceptics. When you have a private fortune, you can afford to play politics. But I won't stand by and let them gamble with people's jobs.
"The fact is that the Conservative Party cannot speak with one voice on Europe and cannot be trusted to lead this important national debate. It would be a disaster if Labour was to follow suit. Under my leadership, Labour will step up to the plate, fill this void and fight to secure the national interest."
He added: "There are of course issues that we need to address as part of this debate, particularly on immigration. We need new EU rules to protect the 'going rate' of the skilled workforce and prevent the under-cutting of wages. And we need more financial help for communities most affected by migration to relieve pressure on public services."
Labour's leadership race has been mired in acrimony as the party struggles to cope with a new system that has inadvertently allowed opponents to join up.
Burnham's campaign has warned the outcome faces being open to legal challenge because thousands of Conservatives may have signed up to vote while supporters of Corbyn claim the race is being rigged after being blocked from voting.
But acting leader Harriet Harman has insisted the contest is being run rigorously and the result "will stand".