In a clear sign the gloves are starting to come off on Friday, the Environment Secretary was being targeted ahead of Sunday's televised debate on Channel 4 - which will not feature Mr Johnson.
Mr Stewart said he felt "inspired" and was "absolutely over the moon" to have secured just enough votes to survive the first ballot.
Mr Johnson, a former London mayor and foreign minister, has spent weeks wooing Conservative MPs, staying out of the spotlight with a low-key campaign at odds with his flamboyant publicity stunts of the past.
He did not immediately commit to backing another runner, saying he needed to "talk to all the other candidates about how these values can be best taken forward".
Sir Alan Duncan, the Foreign Office minister, who worked along Mr Johnson in the department but who is supporting Mr Hunt, said: "Boris is a very high octane, high risk politician".
"I'm incredibly grateful for all the support I've received throughout this campaign and I'm really proud we've set the agenda in fighting for values of free enterprise and a free society and trying to make sure the Conservative Party is focused on delivering for people and winning the centre ground of politics", said the West Suffolk MP.
Jeremy Hunt could end up with Mark Harper's 10 votes but, with Harper considered to be on the Brexit fence, Hunt will need to also muster votes from elsewhere to have a chance.
Klay questionable for Game 3 with mild hamstring strain
Thompson is averaging 19.5 points in 18 playoff games, shooting 44 percent from the floor and 41 percent on 3-pointers. Curry leads the Warriors averaging 27.3 points and has added 5.4 rebounds and 5.2 assists per game.
Trump reverses course, says 'of course' he would report foreign interference
A top Republican ally said Trump was wrong to say he would accept dirt on political rivals from foreign sources. Trump called his former counsel a liar.
They're Playing The Happy Blues In St. Louis
Louis Blues goaltender Jordan Binnington can play this offseason, and general manager Doug Armstrong knows it. HOCHMAN: So I'm chatting with her, and I say, maybe I'll see you on the ice after the game.
The frontrunner now says he will join a TV debate with his rivals once more of them have been knocked out of the contest on Tuesday.
"I think that shows a certain ruthlessness in forcing that through". Mr Stewart was pacing the corridor, hoping to catch colleagues before they voted with one last plea to help get him over the line.
Johnson's spokesman has said he was in "discussions" with broadcasters about participating in their debates. As the face of the pro-Brexit campaign in 2016, he has called for a clean, quick break with the European Union, resigning from Prime Minister Theresa May's cabinet a year ago in protest at her plan to retain the bloc's trade rules. But some of those running to lead the Conservatives, including Mr Johnson, say it may be the only way for Britain to leave the bloc without further delay.
The remaining two MPs will then appear on a final ballot, voted on by general members of the Conservative party.
In his campaign, on the issue of Brexit, Mr Hancock vowed to go to Brussels to broker a time limit to the controversial Irish backstop and said MPs would block a no-deal exit.
Some candidates, including former Brexit secretary Dominic Raab, have even spoken about the possibility of shutting parliament (known as proroguing) in the build-up to Brexit in order to ensure that the United Kingdom does indeed leave on the 31st of October as planned.