Both votes are victories for Prime Minister Theresa May, who only the day before fended off lawmakers who wanted to give Parliament the power to force the government to renegotiate Brexit if it returns with a deal they reject.
The votes in the House of Commons, the elected chamber, could influence negotiations over the future relationship between the European Union and the United Kingdom after Brexit. Currently, the government is offering lawmakers a "take it or leave it" vote on the final deal.
He said: "Facing the prospect of a humiliating defeat, Theresa May has been forced to enter negotiations with her backbenchers and offer a so-called concession".
Solicitor General Robert Buckland said the government remained "open-minded", but this may or may not result in it coming forward with new proposals in the coming days.
This evening, I have resigned as Shadow Defence PPS to vote in favour of the Lords EEA amendment.
Attempts to keep the United Kingdom in the European Economic Area after Brexit have been defeated in the House of Commons, amid a major Labour revolt over the issue.
Even before debate got underway, Brexit sparked drama in the House of Commons Wednesday, with Scottish National Party lawmakers walking out to protest the short amount of time given to debate Scotland-related issues on Tuesday - 20 minutes out of a six-hour session.
This change sought to give greater powers to the "sifting committee", which would be established to decide whether recommendations proposed by ministers to amend retained European Union law after Brexit would require a Commons vote.
Before the vote, she assured lawmakers she would honour her promise and deal with the "concerns raised about the role of parliament in relation to the Brexit process".
The Lords amendment would have given parliament the power to decide what happened next, with the possibility of reopening negotiations or staying in the bloc.
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Mr Trump also responded by announcing that he was backing out of a joint communiqué on trade between the USA and its main allies. Trump went on to say that the U.S. gets "unfairly clobbered" on trade despite "protecting Europe at great financial loss".
Pro-European Conservative, Nicky Morgan, told the BBC that May understands that "parliament wants to have a real say, in all circumstances".
Ms. May's minority government relies on the support of Northern Ireland's Democratic Unionist Party for a slender working majority in the 650-member Commons.
She added: "I can not countenance parliament being able to overturn the will of the British people".
But, to the disappointment of some of pro-European MPs, Downing Street rejected a second plan that would allow parliament to step in if no deal were reached by February.
Eurosceptics urged the government to hold firm, saying the rebels were seeking to undermine the 2016 referendum vote for Brexit.
"Further amendments to the Brexit bill will be discussed today; but even after that I still do not see a clear path that will satisfy both sides of the party, the Brexiteers and the Remainers, sufficiently so as not to topple the government and that will also seem acceptable to the European Union", says Esther Maria Reichelt, an analyst at Commerzbank.
It was always doomed to fail because it was opposed by the Labour party.
In a point of order Ian Blackford, the SNP's leader at Westminster, says Labour has let down the people of Scotland.
He had sought to disrupt proceedings in protest at the government's failure to leave enough time to debate how the legislation would affect devolution.