The ex-Brexit secretary said the party's woeful performance in the European elections and Peterborough by-election underlined the damage already inflicted on the Conservatives.
The stark warning from Mr Raab came as he prepared to participate in a televised debate with four of his rivals to replace Theresa May in Downing Street on Sunday evening.
Clear favourite to succeed the prime minister, Boris Johnson, has refused to attend the Channel 4 debate, and the broadcaster plans to present viewers with an empty podium.
"When people voted, they vote to Leave. We haven't left yet and that's why we're seeing not just the uncertainty for the economy – but also this corrosion of public trust," Mr Raab told Sky News' Sophy Ridge on Sunday.
"The Tory Party will be toast unless we're out by the end of October. People need to wake up this. I certainly think the Conservatives cannot win an election unless we deliver Brexit."
Mr Raab, who claimed his "underdog" campaign is "building momentum", added: "This race has only just started. I appreciate there is a frontrunner here but I am looking forward to this race."
The former cabinet minister also insisted the other candidates had failed to set out a "credible plan" for Britain's exit from the EU.
"I'm not quite clear what Boris's plan is, but in fairness I'll let him set that out in more detail," he said.