If Mrs May loses the vote, the government will be forced to ensure Britain remains in a customs union with the EU if no agreement on trade is reached by January 2019.
The narrow victory is May's third this week, underlining the difficulty she faces in passing legislation on one of the most divisive and important decisions in modern British history with only a minority government and a party at war with itself.
Theresa May scraped through a crunch House of Commons vote after a Tory Remainer rebellion failed to alter her post-Brexit customs plans.
With the prime minister failing to gain universal backing among Conservatives for the Brexit deal she thrashed out with her cabinet at Chequers earlier this month, and facing further rebellions over key legislation, 74% of people say they do not think she is able to keep her MPs in line.
"We have a prime minister who is in office, but not in power", he said. A total of 48 MPs would have to write letters to the Conservative Party's 1922 committee calling for a confidence vote in order for one to be triggered, May would then need to win the support of over half of the party's 316 MPs.
Labour is understood to have instructed its MPs to vote against the move and several pro-European Tories have indicated they will oppose it.
Under the deal, Ms Swinson claimed Conservative Party chair Brandon Lewis had agreed to also sit out the votes, but he was shown on a division list to have voted on the crucial customs union amendment. Eight officials have quit since that summit, including two very high-profile Leave advocates, David Davis and Boris Johnson.
In a series of one-to-one meetings, ministers had tried to appeal to the rebels' "common sense", pointing out that leaving the Customs Union was a manifesto commitment.
"Labour members and voters in Vauxhall deserve better than someone who will prop up the party of austerity, just so that she can help take this country off a cliff edge".
With Brussels expected to reject the government's proposals in the White Paper when Brexit talks resume next week, speculation is mounting that the United Kingdom could face a Brussels ultimatum to choose between a Norway-style deal inside the single market, or leave the European Union with no deal at all.
The first minister also described the UK's Brexit plan, published last week, as a "vaguely credible" negotiating position.
"However, I believe it is a basis on which more serious negotiations can now start".
Although far from lavishing praise on the United Kingdom government's new Brexit plan, it's an interesting change of emphasis from a first minister who has to date exclusively fired his rhetorical arrows towards Downing Street.