The Labour leader said the vote by the parliamentary Labour Party had no “constitutional legitimacy” under party rules.
Although there was no official announcement of the voting figures, sources said it was 172 to 40 in support of the motion, with four spoilt papers.
In a statement, Mr Corbyn said: “I was democratically elected leader of our party for a new kind of politics by 60% of Labour members and supporters, and I will not betray them by resigning. Today’s vote by MPs has no constitutional legitimacy.
“We are a democratic party, with a clear constitution. Our people need Labour party members, trade unionists and MPs to unite behind my leadership at a critical time for our country.”
His determination to fight on — despite the vote and the walk-out of dozens of shadow ministers — means Labour rebels will have to mount a formal leadership challenge if they want to oust him.
Speculation has been rife that deputy leader Tom Watson or former shadow business secretary Angela Eagle — who quit the shadow cabinet on Monday — could run as a “unity” candidate.
Mr Corbyn’s supporters are confident he will win out in a ballot of grassroots activists who swept him to the leadership last year and who will decide the outcome of any contest.
His team insist that if there is a challenge, he will automatically be on the ballot paper as the incumbent party leader.
But some in the party have argued that under party rules he will need the nominations of 50 MPs and MEPs in which case he could struggle to get the necessary support.
Unite general secretary Len McCluskey said: “The extraordinary behaviour of Labour MPs has achieved nothing beyond diverting attention from a Tory government in crisis.
“If anyone wants to change the Labour leadership, they must do it openly and democratically through an election, not through resignations and pointless posturing. If there has to be such an election, Jeremy Corbyn’s supporters throughout the movement will be ready for it.”
A Labour source loyal to Mr Corbyn insisted “our support is still strong” and any attempt to oust the leader would amount to “shoving two fingers up to democracy”.
Rebels who refused to accept Mr Corbyn and shadow chancellor John McDonnell were compared to people who still believed the Earth was flat.
“There are a lot of flat earthers out there who have got to come to terms with the fact the world is not as they like it,” the source said.
The source poured scorn on the prospect of a challenge from Ms Eagle, saying she would be “the ideal candidate for us” because of her voting record on issues such as the air strikes against Syria.