Mr Johnson also snubbed a hustings for political journalists yesterday, though he is due to appear in tonight's BBC TV debate.
The final five candidates in the Conservative leadership contest had their say in a lively televised debate this evening.
But the 45-year-old bagged 37 votes, more than the Home Secretary Sajid Javid who gained just 33 and Dominic Raab who was eliminated with 30.
Tory peer and polling expert Lord Hayward has said there is a "striking antipathy" towards the former Foreign Secretary in traditional Tory areas like the Home Counties.
The former diplomat was only appointed a Cabinet minister last month but has taken a different approach in his bid to become leader, walking across the country meeting people in the street to discuss their issues.
He said that "nobody sensible" in Parliament wanted a general election soon, with the other candidates largely agreeing Brexit needed to be delivered before another public vote.
Asked why that candidate should be Mr Johnson, he said: "When Boris was London mayor he proved he could be a moderate, socially liberal, one nation Tory and he is someone who can get Brexit done".
Jeremy Hunt also made it through to the next round with 46 votes, as did Michael Gove with 41 and Sajid Javid with the bare minimum 33.
All of the party's 313 MPs can take part in the secret ballot in the contest to replace outgoing leader Theresa May, with further rounds scheduled to whittle the list down to a final two, who will face the party's 160,000 grassroots members.
His rivals have stepped up their calls on Johnson to spell out his plans for Brexit in more detail.
Stewart, who started the campaign as a rank outsider but has electrified the race, accused Johnson of selling "fairy tales" about how he would solve the Brexit puzzle.
The four candidates without Stewart said they would be willing to lead Britain out of the European Union without a divorce deal on October 31 - the current deadline set by the EU.
Johnson says he will take Britain out of the European Union by October 31 whether or not there is a deal with Brussels to smooth the transition, potentially setting up a fight with parliament. "But I try my best to connect and I think it's very important as a modern Conservative Party that we reach out to those modern audiences", he told Today.
The finalists will face the ruling party's 160,000 grassroots members in a vote next month to decide the victor. But parliament has indicated it will try to stop a no-deal Brexit, which investors warn could roil markets and shock the world economy, while the European Union has said it will not renegotiate the Withdrawal Agreement that May agreed.
But he has attracted criticism from his rivals for promising to cut income tax for high earners.
A question from Abdullah, an imam from Bristol, led on to a discussion about Islamophobia and Boris Johnson said he was "sorry for the offence" his comments about veiled Muslim women looking like "letter boxes" and "bank robbers" had caused.