Parliament rejected an amendment by Normanton, Pontefract and Castleford MP Yvette Cooper this week to avoid a no deal Brexit.
Ms Cooper’s proposal would have meant Prime Minister Theresa May was told to go back to the EU and ask for an extension if no agreement had been reached on a deal by March 29 – the day the UK is due to leave the EU.
A no deal Brexit could affect supplies of food and medicine.
MPs instead voted for another amendment in the House of Commons on Tuesday evening.
The Brady amendment means that Theresa May will go back to Brussels and demand changes to the Irish backstop and ask for “alternative arrangements”.
It is not clear what those alternative arrangements would be. Within minutes of the Commons agreement the EU said the deal was not open for renegotiation.
The bloc’s chief negotiator Michel Barnier said the other 27 EU nations were “united” over the existing Brexit deal negotiated by Theresa May, which has been rejected by parliament.
MPs voted 317 to 301 in favour of changing the backstop plan – which is the part of the deal designed to avoid the return of Northern Ireland border checks.
In Parliament, Theresa May said: “The world knows what this house does not want. Today we need to send an emphatic message about what we do want.”
Following the vote, Ms Cooper said: “I want the Government to get a workable Brexit deal, but I’m really worried that the Prime Minister’s drift and chaos could mean we end up with No Deal instead even though that would hit medical supplies for our hospitals, put tariffs on food for hard pressed families and cause delays for local manufacturers.
"I’m glad that last night MPs voted against a no-deal Brexit, but we did not get enough support to allow the Government a couple of extra months to get a deal in place if we run out of time.
"The trouble is that the Government has left everything so late that they could end up hurting families and businesses round here and that’s not fair.
"The Prime Minister will end up asking for more time at the last minute and then it could be too late. The Government should get its act together, sort it out and be honest with people."
Fourteen Labour MPs voted with the Government against Ms Cooper’s amendment, which was ultimately defeated by 321 votes to 298.
Their votes helped to cancel out the 17 Tory rebels, who voted for Ms Cooper’s amendment.
Hemsworth MP Jon Trickett and Wakefield MP Mary Creagh voted in favour of the amendment. The value of the pound fell after the decision as the Prime Minister did not rule out a no deal Brexit.