The results could also deliver a further blow to Prime Minister Theresa May's authority, which was severely damaged when she.
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.
Overall MPs voted against the House of Lords' amendments in three separate votes.
As the vote approached, Tory MPs intending to defy the party whip in order to vote with Labour on the amendment left the chamber.
The amendment would effectively give MPs the power to prevent the United Kingdom from opting to crash out of the European Union without a Brexit deal and is a major victory for Remain-supporting MPs in parliament.
If agreed, ministers would have until the end of November this year to secure a Brexit deal before seeking the approval of parliament.
MPs will on Wednesday debate a series of other amendments to the bill, which if passed, would seek to force May to change course and negotiate to stay inside a customs union and the single market after Brexit.
After 11 straight votes on the UK Government's EU Withdrawal Bill, they took to Twitter to vent their understandable frustration at the lack of electronic voting.
Justice minister Phillip Lee is the first minister to resign over the government's Brexit policy.
The United Kingdom is now part of the European Union single market but if London leaves it after Brexit, Britain will have to negotiate new trade deals with its partners, including the United States.
"We have listened wherever possible to sensible suggestions that have been made to improve the bill", he said.
It is thought that both the government and the Remainer group of Tory MPs consider the outcome to be a success. Grieve suggested that would not be enough to prevent a rebellion.
Earlier, most of the original 11 rebels on the meaningful vote amendment had indicated that they would stand firm.
The final vote, as outlined in Grieve third point, would be different as the government would then have to follow any direction given by the Commons.
The major last-minute concession by ministers means the government avoids defeat by 324 to 298.
However, a compromise appeared in sight last night after Remainers and Brexiteers backed a call for ministers to report on attempts to agree a customs "arrangement".