They subsequently warned they will not be easily bought off, while Tory Brexiteers urged ministers not to concede any ground.
Buckland intimated the government had not fully committed to any part of Grieve's amendment.
Brexit campaigners still voiced concern that the concession may open the door to the European Union trying to force Britain into retaining the closest possible ties with the bloc by weakening the government's hand in the talks.
The government's eleventh hour amendment, lifted in large parts from Tory Remainer Dominic Grieve's own proposal and to be presented to the Lords on Monday, is expected to give MPs the right to veto the government's strategy if it fails to secure a political agreement with the European Union by 30 November.
May's fragile government will be trying to defeat a rebellion by pro-EU lawmakers and reverse changes to its key piece of Brexit legislation as the matter comes before the House of Commons for two days of debate. Mr Grieve's proposal will now be added to the legislation. Although, as things stand, they will not be able to send the government back into negotiations if they reject an agreement with the EU.
A Downing Street source said: "We will get a good Brexit deal that works for everybody in the UK".
"We have not, and will not, agree to the House of Commons binding the government's hands in the negotiations". The two sides aren't yet clear exactly on the terms of their deal, but it looks likely that Parliament will emerge with a greater say in the process.
"The government have conceded that this is necessary and I expect to see a new amendment to cover this situation soon".
In a day of drama, May's position seemed suddenly weaker when junior justice minister Phillip Lee, who has always been critical of the government's Brexit strategy, resigned and said he would vote against the government.More news: E3 2018: The Elder Scrolls returns to mobile with ambitious RPG Blades
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Dominic Grieve tabled an amendment last night which forced the government's hand over the issue of a meaningful vote.
"This justifies my decision to resign and makes it a lot less painful".
MPs were told that one parliamentarian had to be accompanied to a public meeting by a six armed police officers because of threats over their stance on Brexit.
It is thought that both the government and the Remainer group of Tory MPs consider the outcome to be a success.
"We are asking members of parliament to abide by the referendum result, our manifesto commitment and to back our country", Andrew Bridgen, Conservative lawmaker and Brexit campaigner, told Reuters.
Dr Lee's shock departure came as Brexit Secretary David Davis warned potential Tory rebels that they can not undo the European Union referendum, ahead of a tricky 48 hours in which the Government will try to get its Brexit programme back on track.
The government says the changes would weaken Britain's negotiating position and is seeking to reverse them in the Commons. Pro-Brexit members of the government want to be able to play the "no deal" card, but the House of Commons, where pro-EU voices are stronger, would nearly certainly reject the idea.
British Prime Minister Theresa May narrowly avoided a major blow to her Brexit strategy Tuesday after MPs rejected a plan that would have given parliament a veto on the final deal negotiated with Brussels.
Ministers and other lawmakers have been told to make sure they will be in parliament for the votes.