Mrs May says Brexit could be delayed by three months, to 30 June, if MPs back her deal in a vote next week.
Lawmakers approved by almost 200 votes a motion setting out the option to have a short delay by agreeing to a Brexit deal by March 20, or a longer delay if no deal can be agreed in time. Another will try to prevent May bringing her European Union twice-rejected divorce deal back for a third vote.
Work and Pensions Minister Sarah Newton quit after defying the whips to vote for the cross-party proposal.
MPs have backed a delay to Article 50 on a third night of votes on Brexit in the House of Commons. Party leader Jeremy Corbyn said after Thursday's votes that a new Brexit referendum might offer a realistic way to break the deadlock.
Postponing Brexit gives May some respite, amid a crisis that has shredded her authority and obliterated her control of a fractious Conservative minority government.
"I implore him not to stand against the amendment today or I'm afraid that Labour will be found out for what they are - a fraud and they are participating in Brexit happening if they fail to back the People's Vote this afternoon".
He said he had canvassed external Brexit campaign groups to find their opinion on whether it was right to vote down the deal.
If no deal was agreed by March 20, "then it is highly likely the European Council at its meeting the following day would require a clear objective for any extension, not least to determine its length, and any extension beyond 30 June 2019 would require the United Kingdom to hold European Parliament elections in May 2019", the motion said.More news: Cavs coach Drew condemns Chriss-Ibaka scrap
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While this marked a smaller defeat than the 230-vote margin with which they rejected her deal back in January, it was a decisive rejection of government strategy even after May had claimed significant changes to the controversial Irish backstop clause had been achieved.
A new round of Brexit tussles came a day after chaotic scenes in the House of Commons, when lawmakers voted to rule out leaving the European Union without an agreement on divorce terms.
UK Prime Minister Theresa May enjoyed a rare good day in Parliament, fighting off her opponents and winning the endorsement of British politicians to seek to delay Brexit day.
May, visibly exhausted and struggling to speak as a result of a sore throat, admitted that Tuesday's vote confirms that there is no support for "this deal" and how the Brexit process moves forward remains all the more uncertain.
On Thursday, President of the European Council Donald Tusk took to social media to reveal that he will be appealing to the EU27 members to remain open to the idea of offering the United Kingdom a long extension period, if they need time to rethink their Brexit strategy.
Some lawmakers have been pressing for a series of votes in Parliament on different Brexit options - including a closer relationship with the bloc than the government wants - to see if any can command a majority.
"Our "no-deal" preparations are now more important than ever before", he added. It is now up to May to go to Europe and see if she can agree to a delay - a delay she previously said would be short-term and no later than June. The motion is not legally binding and does not rule out Britain leaving the EU.