Some Eurosceptic Tory MPs are particularly anxious about an extension as they fear this could end with May reaching out to Labour to work on a softer Brexit deal they could vote for, a second referendum or general election resulting in a Labour victory.
Brexit-backing Conservative lawmaker Edward Leigh said colleagues should back the agreement because "the choice is no longer perhaps between an imperfect deal and no deal - it is between an imperfect deal and no Brexit".
The House of Commons emphatically rejected his Brexit vision by a margin of 240 votes to 323.
Mr Corbyn indicated this week that he would move towards advocating a fresh referendum if Labour's preferred option was rejected by the Commons.
Shadow Brexit secretary Sir Keir Starmer said he was "disappointed" at the defeat of Labour's amendment, adding: "We must do all we can to protect jobs and the economy".
Cross-party supporters of another public vote on Brexit have also put forward an amendment to kick-start that process next month.
Mr Corbyn told Labour MPs on Monday that he was ready to support moves to demand a second public vote "to prevent a damaging Tory Brexit being forced on the country".
MPs voted by 502 to 20 to endorse May's plan, a radical shift in strategy made only after the threat of revolt by ministers who fear Britain crashing out of the European Union on March 29 without a deal.
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Pincered between the two feuding wings of her party, the PM reluctantly ordered all her MPs to agree to the demand in a symbolic vote on Wednesday night.
Parliament will vote on any revised deal by March 12, and if lawmakers again refuse to back it, they would then vote the next day on whether Britain should leave without a deal.
The wire service also reports French diplomats said the nation could only support the Brexit delay British Members of Parliament are likely to vote for if it came with a "credible solution" to the ongoing indecision among the Westminster political elite about how to deal with the withdrawal from the European Union that the British people voted for in 2016. Writing in the Daily Mail, Mrs May said talks with the European Union about securing changes to the divorce deal to make it more palatable to Parliament have "begun to bear fruit".
An amendment tabled by Anna Soubry of the new Independent Group calling of a second referendum was not selected.
French President Emmanuel Macron, campaigning for elections to the EU parliament in May on a platform of deeper and faster sharing of national powers in the Union, echoed Barnier and went further, ruling out granting London more time unless it meant a clear prospect of doing more than just prolonging the agony. "The crisis is now so great the public must be heard".
May is hoping to win round more hardline Eurosceptics to back her deal, playing on their fears that a delay to Brexit could mean a second referendum. All other Liberal Democrat MPs abstained.
An amendment from Scottish National Party MP Ian Blackford that the United Kingdom should not leave the European Union under any circumstances without a deal "regardless of any exit date".
Tory MP Alberto Costa's amendment to seek a treaty on citizens' rights after Brexit was approved without a formal vote.