Saving lives this winter would become a “moral mission” if Labour wins the December 12 election, the party said, as it announced a multibillion-pound package to tackle the crisis.
Rough sleeping has more than doubled since 2010 while the number of people dying while homeless increased to 726 in England and Wales last year, according to official figures.
All three main parties have pledged to end rough sleeping by the end of the next Parliament, but their means differ.
Among Labour’s plans, which it believes are the boldest in 20 years, the party would build thousands of new homes for a “housing first” model to get people off the streets regardless of their situation.
Shadow housing secretary John Healey said a "Labour government would end homelessness" within one parliament.
Over five years, Labour proposes a £600 million “modern hostels fund” to provide quality homeless accommodation, with 9,000 extra bed spaces and a £200 million fund to upgrade existing hostels.
In yearly spending, the party would use £100 million for emergency winter shelter and £1 billion earmarked from council budgets to pay for staffing, support and funding to relink local housing allowance with local rents.
Labour would also expand the “housing first” model nationwide, with 4,000 new homes built to get people in accommodation so they can then begin to rebuild their lives with access to any services or treatment they need.
A further 4,000 permanent “move-on” homes would be built for rough sleepers moving out of hostel accommodation.
Labour said cash for the 8,000 new homes would come out of its £150 billion social transformation fund.
The scale of the crisis was highlighted in a Shelter report this week that suggested 135,000 children in Britain will be homeless and living in temporary accommodation this Christmas.
Labour leader Mr Corbyn said: “One person sleeping rough is one too many.
“No-one wants to live in a society where thousands of homeless people are left out in the cold on the streets.
“Labour will save lives this winter and end rough sleeping within five years. That’s real change."
Sajid Javid, a former Conservative housing minister and now the Chancellor of the Exchequer, told ITV News: "I want us to do even more on homelessness."
Responding to whether he was "ashamed" of his party's record on rough sleeping, which official figures show has risen by 185% since 2010, Mr Javid said: "Homelessness reached its peak in 2003 under the last Labour government and since then it's down by half."
He went on to say the Conservative party "accepts there is more to do" to tackle the issue. The party has pledge £1.2 billion until April 2020 with a further £422 million for 2020-21 to tackle the problem.
The Liberal Democrats would introduce a duty on local authorities to provide immediate emergency accommodation and end no-fault evictions under the party’s plans to end rough sleeping.