They reassured anti-Brexit MPs that the government would accept some of their core demands to give parliament a meaningful say on the terms of Britain's European Union divorce, including - potentially - a new deadline for a deal to be agreed with Brussels that could make it hard for the government.
Earlier, Mrs May was hit by the resignation of justice minister Phillip Lee, who quit the Government live on stage during a speech in London in order to be able to back Mr Grieve's amendment.
On Wednesday, May had faced the prospect of losing a vote after rebels had indicated their support for a change introduced by the House of Lords to require ministers to report what efforts they had made to secure a customs union.
Some lawmakers tried to shout him down and accused the government of wanting too much power.
In a highly charged atmosphere in parliament, lawmakers who oppose the government said they had received death threats and brandished a copy of one of Britain's tabloid newspapers, the Daily Express, which ran a headline saying: "Ignore the will of the people at your peril".More news: Volvo launches Polestar Engineered performance pack
Ukip leader Gerard Batten said: "The only "meaningful vote" was the verdict of the people in referendum of June 23 2016".
But that vote required last-minute concessions to pro-European Tories, and they warned Wednesday they could yet seek to defeat the government if May fails to fulfil her promises.
Ms Cooper added she believes there is a majority across the country and Government in favour of a "close economic relationship", which means some version of single market participation - or as close as the United Kingdom can get to it.
The Commission has expressed its concerns, including its position that London's proposal is overly complex and hard to implement, through the EU's chief Brexit negotiator, Michel Barnier.
In the third major vote of the night, MPs voted on an amendment from the House of Lords aimed at preventing the repeal of the 1972 British Act of Parliament that took Britain into what was then known as the European Common Market.
He said Mrs May had promised to table a fresh amendment based on his own proposal for Parliament to be consulted on the way forward if no deal is agreed by the end of November.More news: Rebel Wilson damages slashed on appeal
Eloise Todd, chief executive of Best for Britain, said: "We have been speaking both to government ministers and opposition MPs to make our case and lobby them on Brexit".
May said her government and the European Union were still working towards an October deadline in talks with the European Union to secure an agreement on the terms of Britain's withdrawal and an outline of the future partnership.
Details of the government's commitment will have to be formalized next week in a new amendment to the bill.
Or as some nervous Conservative Brexiteers are now calling it - "a Brexit in name only".
Facing the prospect of losing a vote on a crucial amendment to the government's flagship Brexit legislation - which was created to empower parliament to vote down the final deal without risking a "no-deal" exit from the bloc - ministers intervened with a concession at the 11th hour even as MPs were wrapping up debate on the controversial measure. The vote came on the first of two days of high-stakes debate and votes in the House of Commons on the government's flagship Brexit bill.
The comments come as Mrs May's Brexit "war cabinet", which includes Leave campaigners Mr Johnson and Mr Gove, thrash out two plans for a customs relationship with the European Union post-Brexit.More news: Black Ops 3 is a bonus free game on PlayStation Plus
If May is defeated in the House of Commons it will be yet another blow to a prime minister whose authority has been challenged several times since last year's election.