In a second round of voting among Tory MPs yesterday, the party's young pretender continued his remarkable rise by topping the poll, winning 90 votes, up from 56 in the first round. Mr Davis won 57 (down five) while Dr Fox came bottom with 51 (up nine) and was eliminated.
Mr Cameron and Mr Davis now go head-to-head in a ballot of the party's 300,000 members, the result of which will be declared on 6 December.
Although Mr Cameron's momentum looks unstoppable, he will be tested by the more experienced Mr Davis during the six-week campaign that the two contenders begin today.
There were early signs that the contest could get dirty, as one Davis supporter raised the prospect of media revelations about the personal life of Mr Cameron, who has been dogged by questions over whether he took drugs as a student.
Sir Nicholas Winterton, MP for Macclesfield, said: "The media is extremely adept at finding out stories and discovering skeletons in the cupboards. Do I think the next six weeks will reveal certain things? The answer is yes."
He added: "We want to give the golden boy [Mr Cameron] a real run for his money. He has had plenty of that [for his campaign] while the other candidates have not."
Last night Mr Cameron was asked on Channel 4 News whether he had used cocaine since becoming an MP four years ago: He replied: "I have absolutely answered your question. I have just said No." Although the Davis camp has been shaken by his unexpectedly tough struggle to reach the shortlist of two, they took heart from Mr Cameron's failure to win the support of more than half of Tory MPs. They said that showed there was a natural right-of-centre majority in the party which could yet deliver victory for Mr Davis, the right-wing standard bearer.
David Wilshire, MP for Spelthorne and another Davis backer, said: "The fact that fewer than half of us voted for David Cameron tells us something. If we can catch the mood of the party here, we can catch the mood of the members. We have a case, we have a cause and we fight on."
If Mr Cameron had won the votes of more than 100 of the 198 Tory MPs, there would have been pressure on Mr Davis to withdraw, allowing the 39-year-old shadow Education Secretary to be crowned party leader immediately.
But Mr Davis was quick to scotch the idea after last night's result was announced, blaming speculation that he might quit the race as "black arts" by his rivals. Damian Green, who backed Mr Davis, said rumours of a Cameron coronation were "a classic dirty tricks operation".
In turn, members of the Fox camp put their defeat down to tactical voting by Cameron supporters to ensure their man had a run-off against Mr Davis because he feared Dr Fox would be a stronger opponent. "We know that they were voting Davis to stop Liam," said one member of the Fox team. Dr Fox, the shadow Foreign Secretary, rejected calls from right-wing MPs for him to forge a "Thatcherite" dream ticket with Mr Davis to overcome Mr Cameron. Supporters of Dr Fox served notice they will be seeking a right-wing Eurosceptic agenda from the two remaining runners.
Gerald Howarth, a member of the Fox team, said: "The non-Cameron camp did command 106 votes against David Cameron's 90. It's not a shoo-in and I think this debate is actually what we need. It is what David Cameron needs. What we have now seen is that we have a man who can conduct himself in a fluent manner, who has ideas, who can withstand some pretty virulent attacks in the press. That is a quality one should not downplay."
Dr Fox declined to say which candidate he would support, saying he would be watching them during the next six weeks. "I want to endorse the programme I have set out in the past few weeks," he said. Although he was "disappointed" not to have made it through to the final round, he said: "We came from behind all the way through and provided everyone with a great run for their money."
The jubilant Cameron team, who surrounded the winner outside the Commons, said they were confident Mr Cameron would carry the party in the country on the bandwagon that began rolling only two weeks ago at the Tory conference in Blackpool.
"I am amazed by the shift of opinion in the constituencies," said a former Tory minister. "There has been a surge of support for David Cameron among my women members who liked what they saw at the party conference. I have never known a shift of opinion like it. Even the drugs issue didn't matter. They said to me, 'Cocaine, so what? It was all a long time ago'." Mr Cameron, who planned to spend a quiet night at home celebrating over dinner, will come under strong pressure to deliver more details of his policies to answer criticism that he is all presentation and no substance.
Saying he was "delighted" with the result, Mr Cameron added: "I'm looking forward to taking my message right round the country. It's going to be a lot of hard work, but I am very excited by it. I want a party that looks to the future, a party that is a 21st-century party and is modern and compassionate and understands the aspirations and hopes and dreams of the people."
His campaign manager, the shadow Chancellor George Osborne, said "It's a tremendous result, exactly in the range we were predicting. David Davis has dropped votes. This is a contest we are relishing."
Outside Parliament, Mr Davis said: "There's a long time to go in this contest yet, still another six weeks. I intend to fight for my beliefs."
Momentum gathers for the 'Tory Blair' campaign
David Cameron enters the run-off with a strong mandate from the party's MPs. His dramatic progress from outsider to favourite is one of the most remarkable rises in recent times.
At every stage of the contest, the 39-year-old shadow Education Secretary has done better than expected, giving him the crucial commodity of momentum. Yet after Kenneth Clarke entered the race in August, Mr Cameron looked like a boy in a man's game and some allies wondered whether he might withdraw gracefully.
After a smooth campaign launch, the turning point came when Mr Cameron wowed the Tory faithful at the Blackpool conference in a speech made without notes. Although Mr Clarke arguably made a better one, he was not "new" or "fresh"; Mr Cameron was the talk of the town and the undoubted star.
By the end of the week, he was the favourite and his bandwagon continued to roll. His nerve was tested by questions about drugs but he weathered the storm.
Mr Cameron told his team yesterday he wanted the members' ballot to go ahead so he could secure a mandate from the whole party.
Supporters say his emphasis on presentation is right because the Tories need to look and feel like a new organisation. But he will come under pressure over his apparent lack of detailed policies.
Some MPs believe the Cameron bubble could burst and his campaign could implode as quickly as it took off. But most believe he will win a comfortable victory and "Tory Blair" will face Tony Blair at Prime Minister's Questions the following day.
Tough as old boots, but Davis faces uphill battle
He should have been sleepwalking to victory. But having been the favourite for the crown, the Tory leadership contest has turned into an uphill battle for David Davis.
Amid a smear campaign, it was claimed on the eve of the ballot that he would stand down for a "coronation" for David Cameron, should he come a poor second. The rumour was fuelled by his aides when they refused to rule it out. However, it was calculated to boost Dr Fox's support.
Even as his campaign floundered, the ex-SAS man kept his assassin's thin smile. "He's tough as old boots," said an ally.
He was not noted for speeches, but the jitters began after his wooden conference address. It contrasted badly with the eloquence of Kenneth Clarke and Mr Cameron.
His unique selling point was that he was not a typical Tory - he was the son of a single mother, brought up on a tough London council estate, and could reconnect the party with the hoi polloi.
But by yesterday even Tory MPs, who had been attracted to his flag, were turning against him. Some found excuses to vote for Mr Cameron in the privacy of the ballot box and some right-wingers had scores to settle. The Eurosceptic right never forgave him for being the whip in charge of securing a Commons majority for the hated Maastricht Bill. Other right-wingers accused him of plotting Iain Duncan Smith's downfall.
The reputation for strong-arm tactics by the Davis camp became a joke when one of his supporters, Graham Stewart, voted with his arm in a sling. Pointing to his arm, he joked: "It says, 'Derek woz 'ere'."
David Cameron 90
David Davis 57
Liam Fox 51