Shadow Brexit Secretary Keir Starmer commented: "This vote was about ensuring parliament was given a proper role in the Brexit negotiations and that we avoid a no deal situation, which is becoming more likely with the divisions at the heart of this government".
Justice minister Phillip Lee is the first minister to resign over the government's Brexit policy.
Theresa May has averted an embarrassing defeat over the government's EU Withdrawal Bill by giving MPs a say on the next steps for Brexit if she fails to secure a withdrawal agreement by the end of November.
Also on Tuesday, the government successfully overturned an attempt to remove the date of Brexit from the face of the bill.
Solicitor General Robert Buckland promised to work on the proposal taking into account concerns raised by former attorney general Dominic Grieve, a leading pro-European rebel.
A Lords amendment to the EU Withdrawal Bill would have given MPs the decisive say on what happens over Brexit if they don't agree with the final deal with the EU.
Lee said that within government he "found it virtually impossible to help bring sufficient change to the course on which we are bound".
Passions ran high in Tuesday's three-hour debate, when angry eurosceptics accused their rivals of trying to undermine the 2016 referendum vote to leave the EU.
Rebels said May had assured them the government would accept two parts of Grieve's amendment; and immediately open talks on a third, "part c", which would allow MPs to direct the government, if no deal is reached by February next year.
The debate, which lasted for almost three hours, was split down the usual non-partisan lines that have emerged as a result of Brexit, with the likes of Labour's Kate Hoey and John Mann saying they would back the Conservative government, while Tories including Ken Clarke and Anna Soubry spoke in favour of Grieve.
Addressing Conservative backbenchers in Westminster on Monday evening, the Prime Minister warned if a series of Lords amendments to the EU Withdrawal Bill are approved by MPs, it will send the wrong message to Brussels.
If the government avoids defeat, is Brexit a done deal?
The Daily Express, another pro-Brexit tabloid, issued a thinly-veiled threat to lawmakers, saying they should "Ignore the will of the people at [their] peril".
The amendment in question hinges on whether lawmakers will get a "meaningful" vote on Britain's membership of the European Union.
Dismissing claims that Mrs May had effectively abandoned her threat that the United Kingdom could leave the European Union without a deal, Mr Jenkin said: "There is only agreement for discussions, not concessions".
Davis on Tuesday told the BBC, "that was the decision of the British people. and whatever we do, we're not going to reverse that".
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