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.
Rory Stewart, a cabinet minister vying for the Tory crown, also attacked Mr Johnson, claiming his plans for Brexit will come "off the rails" once subjected to a detailed examination.
"How is Boris going to deliver Brexit? He keeps saying 'I am going to deliver it'. I don't even know what he believes. He won't talk to me. He won't talk to you. He won't talk to the public," he told BBC1's The Andrew Marr Show.
"We want to know what he believes. The real problem with politics is a problem of trust.
"Nobody has had the chance to question him and as soon as you question him and as soon as I sit down with him and ask the big question - how, how are you going to deliver Brexit? How are you going to get a no-deal through? - Then it begins to come off the rails.
"We are a moderate country, we are not a Trumpian country," Mr Stewart added.
But on Sunday, Mr Johnson's campaign received a boost as Esther McVey, a former cabinet minister who was eliminated from the leadership contest last week, threw her support behind the frontrunner.
She claimed Mr Johnson had agreed to back her agenda for a "blue collar conservatism" if he wins power, adding: "He has promised to deliver Brexit on 31 October, deal or no deal, and has shown time and time again that he is a dynamic leader, capable of building a strong team around him that will deliver on his promises."
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