A motion in the Prime Minister’s name, authorising her to request an extension to the two-year Article 50 negotiation process, was passed by 412 votes to 202 - a majority of 210.
Only a refusal by the leaders of the 27 remaining EU states to grant the UK an extension at a Brussels summit next week could now preserve the totemic date of 29 March as Brexit Day.
Mrs May has made clear that she will press her Agreement to a third “meaningful vote” in the Commons by March 20 in the hope of securing the support of MPs who rejected it by 230 votes in January and 149 earlier this week.
But despite the Prime Minister’s Commons victory, she suffered a further blow to her leadership as eight Cabinet ministers - including the Brexit Secretary Stephen Barclay - voted against the plan, with Chief Whip Julian Smith abstaining.
A majority of her MPs also went against her, with 188 Tories rejecting the motion and only 112 voting in support of Mrs May.
Speaking afterwards, a Number Ten spokesman insisted that the Prime Minister’s top team remained “united” around her Brexit vision and said that Conservative MPs were given a free vote on the motion.
If Mrs May succeeds in getting her deal over the line next week, she will go to Brussels next Thursday to request a short delay to a date no later than June 30, to give herself time to pass legislative changes necessarily for a smooth and orderly Brexit.
But if her deal is rejected for a third time, she believes any extension would have to be far longer and would involve the UK taking part in European Parliament elections in May.
Cabinet Office minister David Lidington told MPs that in this case, the Government would stage two weeks of debate following the March 21-22 summit for the Commons to try to establish a majority around a different plan.
Ministers met for a reportedly testy political meeting of Cabinet ahead of the votes, at which Mrs May was said to have berated four senior colleagues who defied the Tory whip earlier in the week to abstain in the no-deal vote.
European Council president Donald Tusk has indicated that the EU may be ready to offer a lengthy extension to negotiations if the UK wants to “rethink its Brexit strategy and build consensus around it”.
But any extension must be approved unanimously by the EU27, and Austrian foreign minister Karin Kneissl said there could be “some problem” in obtaining this if it took Brexit beyond the date of elections.
Irish foreign minister Simon Coveney told RTE radio: “If you have a long extension of, say, 21 months to the end of 2020 - whatever the period would be - then Britain has a legal entitlement to have representation in the European Parliament” and so must take part in the elections.
Earlier, MPs rejected holding a second referendum on Brexit by 334 votes to 85.