Thirdly, Ms Rudd abstained from voting on the government's amended motion ruling out a no-deal Brexit, which was also passed.
If the bid to take control of Brexit away from her had been passed, the third vote on Mrs May's twice-rejected deal with the European Union would have been her only remaining chance to get it through the Commons. May tries to push her deal through.
MPs, however, also voted against a proposal to hold a second referendum during the delay, dealing a blow to the hopes of pro-EU campaigners.
Labour MP Wes Streeting, who supports a second referendum but voted against the amendment, said that today was not "the right time" to back another vote.
He added that in case there would be no majority for May's deal in the Commons in the third attempt on 20 March, "postponing the withdrawal date only makes sense, if the government and parliament in London change their red lines".
The vote means Theresa May will now head to Brussels next week seeking to extend Article 50 to June 30.
Leader of the Commons Andrea Leadsom said the government's Attorney General, Geoffrey Cox, had considered the matter and would comment further if he thought it was necessary.
Her authority hit an all-time low this week after a series of parliamentary defeats and rebellions, but finance minister Philip Hammond said her plan was back on the agenda. If May's deal doesn't pass next week, she will ask the European Union for a delay of unspecified length, adding another great big unknown to a process already full of them.
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Although the House of Commons on Wednesday voted against the prospect of a no-deal Brexit, the default position if nothing else is agreed remains that Britain will exit without a transition arrangement on March 29, a scenario business leaders warn would bring chaos to markets and supply chains.
Sterling, which swung more wildly this week than at any point since 2017, fell on Thursday from nine-month highs as investors turned cautious about May's chances of getting her Brexit deal approved next week.
"Of course we understand that Brexit is a major issue for each and every one of us, but we would like to have more time to have a debate and allow the public to have more of a say in how to solve the Brexit conundrum".
"I will tell you, I'm surprised at how badly it's all gone from the standpoint of a negotiation", Trump told reporters in the Oval Office as he met Irish Prime Minister Leo Varadkar.
"I think it could have been negotiated in a different manner, frankly". She didn't listen to that, and that's fine.
"I would not read anything in that the chancellor was there other than he is a senior member of the government who is clearly involved in numerous issues we are talking about", Dodds said. Brexit supporters say that, in the longer term, it would let Britain forge trade deals across the world and thrive.
Mr Tusk said: "I will appeal to the EU27 to be open to a long extension if the United Kingdom finds it necessary to rethink its Brexit strategy and build consensus around it".
German Chancellor Angela Merkel and Laos Prime Minister Thongloun Sisoulith hold a news conference at the Chancellery in Berlin, Germany, March 13, 2019.
Lawmaker Andrew Bridgen accused her of pursuing a "scorched earth" policy of destroying all other Brexit options to leave lawmakers with a choice between her deal and a long delay.