Labour said May had been forced to avoid a "humiliating defeat" and "to enter negotiations with her backbenchers".
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.
May objected to the amendment - inserted by the House of Lords - because she said it would tie her hands in the negotiation.
The bill will then go back to the Lords on Monday.
The result followed a day of high drama as a justice minister in May's government resigned in order to back the amendment.
May had earlier warned that defeat would weaken her hand in exit talks, while a string of eurosceptic MPs stood up to accuse the rebels of trying to thwart Brexit.
Talks with Brussels have stalled over the fraught issue of the Irish border, but both sides are hoping to agree a final deal by October in time for the break on March 29, 2019.
It was a message reinforced by her spokesman, who described the bill as "a vital piece of legislation for ensuring our statute book is ready for Brexit day and for delivering the smoothest possible exit from the European Union".
"In due course we will have to tinker around with it in the Lords to improve it". That effectively rules out threatening the European Union with a no-deal Brexit, because the Commons will not approve a no-deal Brexit.
Pro-EU demonstrators hold placards and wave flags as they protest against Brexit, outside of the Houses of Parliament in central London on June 11, 2018. A victory for the "meaningful final vote" amendment would leave the government weaker in am upcoming round of talks with European Union negotiators in late June, and also weaken Theresa May's authority as leader.
Mr Grieve's proposal will now be added to the legislation.
"Whatever we do, we're not going to reverse that (decision to leave the EU)", David told BBC radio.
"The question is how do we take some sensible steps to anticipate that happening and try to make sure that there is a coherent process for dealing with it".
Hours before the debate began, a justice minister resigned in protest at what he called its "wish to limit" the role of parliament in shaping Brexit.
Remain-supporting Conservative Anna Soubry, one of the MPs who met Mrs May on Tuesday, said she trusted Mrs May to "honour the undertaking she gave". The strength of this commitment is yet to be seen in writing - and the Brexit department is still insisting it has not given up control of the negotiations - but the anti-Brexit rebels showed they have the numbers to force a defeat should the government renege on its pledge.
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".
But Tory MP Andrew Bridgen, a leading Brexit backer, said the concessions could "come back to haunt" the government if they amounted to a veto over the terms of the UK's departure.
MPs will vote on Wednesday on whether to dismiss a plan proposed by the upper chamber of parliament which would require ministers to report what efforts they had made to secure a customs union with the EU by the end of October.