MPs on Thursday voted by 412 to 202 for a motion which instructs the prime minister to ask the European Union for an extension to the Article 50 withdrawal process.
If accepted by all the 27 other EU member states, it means that the United Kingdom will not leave the EU on March 29 as originally scheduled.
May has told MPs that if they agree to back a Brexit deal by Wednesday next week, the UK's departure will only have to be delayed by three months, meaning Brexit would take place on June 30.
However, if the House of Commons does not approve a deal by Wednesday, the UK government will be forced to ask for a much longer delay, in order to create time to find a new way forward, the prime minister has said.
Theresa May keeps control of the Brexit process
In other votes on Thursday, the government narrowly defeated a hugely significant amendment which would have given MPs power to control what should happen next if May failed to get her deal through.
The amendment tabled by senior backbench MPs including Labour's Hilary Benn and Yvette Cooper would have paved the way for "indicative votes" on a series of alternatives to May's deal, including a softer Brexit.
The House of Commons rejected it by 314 votes to 312, giving the government a majority of just two votes.
Elsewhere, MPs comprehensively rejected an amendment tabled by the Independent Group's Sarah Wollaston, which called for Brexit to be delayed in order to hold a new referendum, by votes 334 to 85.
Swathes of MPs who support a fresh referendum, or what campaigners call a "People's Vote," did not support the amendment as they believed it was the wrong time to push for it.
The prime minister plans to put her Withdrawal Agreement to a third "meaningful vote" next week, despite the fact it has already being rejected comprehensively on two occasions.
SEE ALSO: The UK will scrap 87% of tariffs and refuse to set up Irish border checks under a no-deal Brexit
