The proposal - put forward by Tory MP Sir Graham Brady - had the support of the government and won by 16 votes.
Theresa May had urged MPs to vote in favour of it, to give her a mandate to return to Brussels and re-open negotiations in order to secure a "legally binding change".
But the EU has said it will not change the legal text agreed with the UK PM.
MPs voted on a string of amendments to Mrs May's plan to change the direction of Brexit.
Mrs May said that, after taking the votes into account and talking to the EU, her revised deal would be brought back to the Commons "as soon as possible" for a second "meaningful vote".
Another amendment, rejecting a no-deal Brexit, also won the support of Parliament on Tuesday, but the vote was not binding - meaning the date for exit remains 29 March.
Labour leader Jeremy Corbyn said as a result of the message from MPs rejecting no deal, he would now meet the prime minister to discuss the next steps. [...]
Five other amendments, including Labour MP Yvette Cooper's bid to delay Brexit if Mrs May does not get her deal through Parliament, were defeated. [...]
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