But other candidates warn that pursuing a no-deal Brexit - the route Johnson has endorsed - would cause economic damage and would fail to pass Parliament.
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.
Pushing himself as the only candidate who could deliver Brexit on time and unite the divided country, he said: "Around the country there is a mood of disillusion, even despair at our ability to get things done".
MPs opposed the motion by 309 votes to 298.
The ruling Conservatives fear that a loss in the next election could bring leftist leader Jeremy Corbyn's Labour party in power.
Several of Mr Johnson's rivals, including Rory Stewart and Matt Hancock, have said they would not countenance leaving the European Union at all without some form of legally-binding agreement because of the economic disruption it would cause. They have been making their pitch to the small electorate of 318 Conservative MPs who will vote in the first of the two-phase election, to be followed by party members voting on final two candidates; the result is due in week beginning July 22.
After Michael Gove's admission that he had taken cocaine in the past, Mr Johnson sidestepped a question as to whether he had also used the drug.
"I think what most people want us to 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 said.
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".
"It is a first and limited step to ensure parliament can not be locked out of the Brexit process over the coming weeks and months", he said.
Sir Bill said: "He has put his name to an opposition day motion with Marxist Jeremy Corbyn".
Opening the debate today Labour's shadow Brexit secretary, Keir Starmer called the measure a safety valve so MPs could begin legislation to stop a new prime minister suspending parliament.
Tory MPs who have said they will back the plan include Antoinette Sandbach, Dominic Grieve, Sam Gyimah, Jonathan Djanogly and the ex-Conservative Nick Boles, who has flown back to the United Kingdom specifically to vote on the motion.
North West Cambridgeshire MP, Conservative Shailesh Vara, voted against the motion, as did Brexit Secretary and North East Cambridgeshire Steve Barclay, South Holland and The Deepings MP John Hayes and Corby MP Tom Pursglove.