The motion to reject the Lords amendment, which would have given MPs control over the government's negotiation strategy and the final exit deal, therefore passed by 324 votes to 298 - a majority of 26 votes.
Shortly before MPs began voting, Sandbach praised the government for making "important concessions".
May is resisting changes approved by the House of Lords that would soften Britain's exit from the European Union, because she says they will weaken the government's negotiating position.
A third development, which would allow MPs to direct the government on future talks if there is no resolution on the withdrawal agreement by 15 February, is also on the cards.
Two of the putative rebels, Anna Soubry and Heidi Allen, quickly said that did not match their recollection of what had been promised, with Soubry urging the government to "sort it please".
The government has accepted one of the Lords amendments, allowing the United Kingdom to continue to co-operate with European Union agencies, which it says is already its policy.
The government was putting a combative spin on the concessions Tuesday evening: "The Brexit Secretary has set out three tests that any new amendment has to meet - not undermining the negotiations, not changing the constitutional role of Parliament and Government in negotiating worldwide treaties, and respecting the referendum result", a spokesperson for the Brexit department said in a statement.
"If, in the future, I am to look my children in the eye and honestly say that I did my best for them I can not, in all good conscience, support how our country's current exit from the European Union looks set to be delivered", he said.
"Where some of its most senior people who hold the greatest offices of state, at every twist and turn, when our Prime Minister moves towards securing a Brexit that will serve everybody in our country, the softest, most sensible Brexit, both publicly and privately they undermine her and scupper her attempts".
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She said the global community was at the beginning of a very long process: "The detail is still to be worked through". So you never know. "So talking about withdrawal of U.S. forces is something that needs to be really thought through".
After PMQs, a No 10 source was asked if that meant, "as far as the government's concerned, clause C is not up for discussion as part of this amendment?" "I'm fairly confident we will be able to do that". Its Lords were instrumental in bringing about some of the amendments that have been sent back to the Commons.
Buckland indicated the government would look into the possibility of adopting Grieve's push for ministers to secure parliamentary approval for their Brexit plans if they fail to negotiate a deal with the EU.
Those developments are likely to lower expectations of a Bank of England interest rate rise in August, given the central bank has said it wants to see more wage pressures before it increases borrowing costs.
Following a meeting with Theresa May, Tory MPs said they had been promised "input" into what the government would do if the United Kingdom faced a no-deal scenario. In three hours, a constitutional absurdity had become a political necessity.
"There's a reasonable expectation that something will emerge, but I need to work on that and more importantly work with colleagues to come up with something".
To avoid a hugely damaging defeat, the rebels said May promised changes would be introduced to the Bill to ensure that MPs get a real say on the final deal.
Sarah Wollaston warned that any government amendment added to the bill when it returns to the House of Lords "must closely reflect Dominic Grieve's amendment".
"The gap is: what happens if there is no deal?".