The vote was followed by the prorogation of Parliament in the early hours of Tuesday, with some Labour MPs holding signs bearing the word "silenced" and shouting "shame on you" as the ceremony unfolded.
Brighton MP Lloyd Russell-Moyle appeared to try to hold on to the Speaker John Bercow at the point he was requested to lead MPs to the Lords as part of the prorogation proceedings.
The prorogation, suspending Parliament for five weeks and ending the longest parliamentary session in UK history, makes a general election extremely unlikely until at least mid-November.
MPs voted 293 to 46, short of the two-thirds majority needed, in favour of a snap election.
The Prime Minister had said he wanted to head to the polls next month to break the political deadlock, as he accused opposition parties of making "outrageous excuses" to delay.
Mr Johnson, who on Tuesday was expected to continue campaigning for the election he is yet to successfully call, said Jeremy Corbyn had become the first leader of the opposition in the country's history to "show his confidence" in the Government "by declining the opportunity to have an election with a view to removing the Government".
But amid furious scenes in the chamber Mr Corbyn said he would not let his party walk into "traps laid by this Prime Minister".
"This Government is only interested in shutting down Parliament to avoid any scrutiny," the Labour leader said.
The PM insisted he would not ask for another Brexit delay, despite royal assent being given to legislation requiring an extension to the UK's EU membership unless a divorce deal is approved or Parliament agrees to leaving the EU without one by October 19.
Mr Johnson said: "It's plain from the turbulent reaction of the benches opposite that they simply want another delay and I will not have that.
"I must warn members that their behaviour in thwarting the will of the people is undermining respect for this House in the country.
"If honourable members want a delay, the only proper way to do it is to ask permission from our masters the people, from our masters the voters."
Following the defeat, Mr Johnson said the Government would "press on with negotiating a deal while preparing to leave without one" ahead of the European Council summit on October 17.
The Prime Minister said: "No matter how many devices this Parliament invents to tie my hands I will try to get an agreement in the national interest.
"This Government will not allow Brexit to be delayed any further. While the opposition run, they cannot hide forever."
In a day of high drama in the Commons in which Mr Bercow announced his intention to stand down next month, MPs also demanded the Government publish communications connected to prorogation and no-deal Brexit planning.
Former attorney general Dominic Grieve's demand for all written and electronic contact about the temporary suspension of Parliament and Operation Yellowhammer documents since July 23 to be released was approved by 311 votes to 302, a majority of nine.
Meanwhile, Mr Johnson will visit a primary school in London to mark the launch of an education drive which could see up to 30 new free schools established.