The government says the changes would weaken Britain's negotiating position and is seeking to reverse them in the Commons.
Tory MP Anna Soubry has told MPs that she knows of at least one Conservative colleague who will not vote with their conscience on the Brexit bill due to fears of violence against themselves, their family, or their staff.
The Bracknell MP, who called for a second referendum on whatever deal Mrs May secures from the European Union, later told the Commons there was growing evidence that the Government's Brexit policy is "detrimental to the people we were elected to serve".
Lawmakers backed a government plan, ending a rebellion that would have challenged May's authority at a time when she is increasingly under pressure to move ahead with all-but stalled Brexit talks in Brussels by offering a more detailed plan.
The PM avoided defeat in a crucial Brexit vote last night after a last-minute concession to Tory rebels over the so-called "meaningful vote" on the final Brexit deal and Dominic Grieve, the former attorney general who led the rebellion, suggested Dr Lee's bombshell resignation triggered her climbdown.
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The Lords amendment on a "meaningful vote" for MPs was defeated in the Commons by 324 to 298.
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 Solicitor General Robert Buckland said while talks may "yield fruit" - he could not guarantee a change in policy. The two then discussed a deal in whispers as other lawmakers made speeches around them.
Shadow Brexit Secretary Keir Starmer commented: "This vote was about ensuring parliament was given a proper role in the Brexit negotiations and that we avoid a no deal situation, which is becoming more likely with the divisions at the heart of this government".
"I'm quite satisfied that we are going to get a meaningful vote on both "deal" and "no deal", Grieve told Sky News.
The most contentious was the bid to give Parliament the power to tell the government what to do if the Brexit deal was voted down or no agreement was reached.
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Conservative MP Sarah Wollaston said on Monday morning she was "minded" to defy party orders and back the meaningful vote amendment.
With only 10 months left before Britain is due to leave, her government is under pressure from European Union negotiators, businesses wanting clarity, and from many in the country to start taking decisions on its preferred future trading ties.
But she faces a gruelling bout of "parliamentary ping-pong" with the Lords, as the Bill bounces back and forth between the two Houses over the coming weeks.
There is little May can do.
Ms. May's minority government relies on the support of Northern Ireland's Democratic Unionist Party for a slender working majority in the 650-member Commons.
Often she simply puts off votes that could end in embarrassing defeats.
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He said: "I'm conscious that if we're to make progress we ought to try and do this by consensus, but he must also understand the difficulty the House is in when it is faced with this kind of choice".