Pro-EU lawmakers later accused the government of going back on its word by offering only a symbolic "take it or leave it" vote on the final deal and not the ability to take control of the negotiations.
The Lib Dems said the "so-called Tory rebels" had "lost their bottle and caved into yet another pathetic government compromise that isn't worth the paper it is written on", while SNP leader Nicola Sturgeon said the concession was a "fudge".
However, although she has promised to give legislators a vote on the terms of the final Brexit agreement negotiated, May insists that, "Her offer to lawmakers, when it comes, will be to take it or leave it".
The Brexit Withdrawal bill will now go back to the House of Lords where it is nearly certain to be supported, paving the way for Royal Assent by Queen Elizabeth II.
Following the Commons vote, the Lords accepted the government's new proposal without a vote.
A victory for the pro-EU rebels would embolden them ahead of debates next month on Britain's future trading relationship with the European Union, which they are seeking to keep as close as possible.
He described Brexit as a national calamity for Britain.
A DExEU spokesman said: "Agreeing to amendable motions would allow Parliament to direct Government on its approach to exiting the European Union and so does not meet the reasonable tests set out by the Prime Minister and Brexit Secretary last week". To cries of "shame" from the Labour benches, Mr Grieve admitted ministers had not given significant ground, but said he had secured a way for the House to make its view on a Brexit deal "clear".
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Peers approved the government's proposal without a vote.
Six Tory MPs - Ken Clarke, Anna Soubry, Sarah Wollaston, Heidi Allen, Antoinette Sandbach and Phillip Lee - rebelled while four Labour MPs backed the government and six other Labour MPs abstained, as did Kelvin Hopkins, who sits as an independent after having the whip withdrawn.
The bill, which returned to the Lords immediately after the Commons vote, is expected be waved through by the upper chamber and could be on the statute book within days.
Labour MP Naz Shah had to be pushed to vote in a wheelchair with a bucket on her lap in case she vomited.
"I think that by honouring that decision, we are showing that in the United Kingdom, in the mother of parliaments, when the people speak their voice is heard".
MPs will then vote on this statement.
"My biggest worry about Brexit is that I don't know what we are planning for", said Juergen Maier, the UK CEO of German engineering giant Siemens.
He said he received an "obvious acknowledgement" of the sovereignty of Parliament and was "prepared to accept the government's difficulty and support it". Other rebels have also reportedly, "called off the revolt", including former cabinet minister Nicky Morgan.