Mr Johnson, the favourite for the top job almost three years after he led the referendum campaign to leave the European Union, praised the strength of the British economy and promised to tackle widespread disillusionment over the protracted Brexit process.
Shadow Brexit minister Jenny Chapman said the vote on blocking a no-deal Brexit was a "safety valve" aimed at Tory leadership contenders trying to freeze MPs out of the decision.
Since Theresa May stepped down as the UK Prime Minister after failing to deliver a favourable Brexit deal, the race to deliver a new deal has gotten tougher.
"Now is the time to unite this country and unite this society", he said, stressing that this task can be only achieved after leaving the EU.
MPs voted 309 to 298 against the measure which would have enabled politicians to size control of the House of Commons later this month.
For many, the contest for prime minister is his to lose - he has the most declared Conservative supporters in parliament and is widely popular among the party's members, the people who will ultimately choose May's successor.
Gareth Snell said he should have backed the agreement when it was put before the Commons for a third time, adding he would support a deal in future.
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Mr Johnson said: "I think what most people in this country want us to really focus on in this campaign, if I may say so, is what we can do for them and what our plans are for this great country of ours".
He acknowledged that his use of language - such as his description of Muslim women who wore the burka as letter boxes - sometimes resulted in "some plaster coming off the ceiling".
British MPs have voted against a Labour Party bid to seize control of the parliamentary agenda from the government in order to try and block a no-deal Brexit.
A party source said: "It provides the first step in the process that can help provide a legislative lock against no-deal on October 31".
At Wednesday's launch, the former foreign secretary - who quit cabinet over Mrs May's Brexit policy - said it was "right for our great country to prepare" for a no-deal outcome.
Commenting on the motion, Rory Stewart said: "For the avoidance of any doubt - I have read the Labour motion proposed for tomorrow and I will NOT be voting for it".
Those two candidates will then face a straight vote by the party's grassroots members, with the victor announced towards the end of July.