In a dramatic day in the Commons, she urged MPs to give her “the clearest possible mandate” to go back to European leaders and restart negotiations.
Parliament later backed an amendment - brought by Tory grandee Sir Graham Brady and supported by the Government - demanding that the contentious Northern Irish backstop, which would create trade barriers between the region and the rest of the UK, was dumped.
The plan, which requires that the backstop is replaced with “alternative arrangements” to avoid a hard border in Ireland, passed by a slim majority of 317-301.
MPs also gave the clearest indication yet that a no-deal Brexit should be ruled out as they backed an amendment from Tory ex-minister Caroline Spelman calling for an agreement to be reached.
Speaking after a crucial day of debate in the House of Commons, Mrs May told MPs: “It is now clear that there is a route that can secure a substantial and sustainable majority in this House for leaving the EU with a deal.
“We will now take this mandate forward and seek to obtain legally binding change to the Withdrawal Agreement that deal with concerns on the backstop, while guaranteeing no return to a hard border between Northern Ireland and Ireland.
“My colleagues and I will talk to the EU about how we address the House’s views.”
But the Prime Minister also admitted that re-opening Brexit talks on the backstop could meet in resistance in Brussels.
She said: “There is limited appetite for such a change in the EU and negotiating it will not be easy but in contrast to a fortnight ago, this House has made it clear what it needs to approve a Withdrawal Agreement.”
Mrs May offered a second invitation to Labour leader Jeremy Corbyn to meet her for Brexit talks, which he agreed to. And she issued a warning to MPs who had registered their dislike of a no-deal Brexit, saying: “Simply opposing a no deal is not enough to stop it.”
The crucial amendment from Sir Graham - chair of the influential Tory backbench 1922 Committee - was one of seven selected by Commons Speaker John Bercow for consideration by MPs.
A cross-party proposal from Normanton, Pontefract and Castleford MP Yvette Cooper and Tory backbencher Nick Boles, which paved the way for delaying the UK’s departure day from the EU, was defeated by 298-321, despite backing from from the Labour leadership.
Jeremy Corbyn’s own amendment - which would have allowed debate on Labour’s preferred Brexit plan or a second referendum was also rejected by 296-327.
Opening the debate earlier in the day, Mrs May told the Commons she would return to the House “as soon as possible” with a revised deal, which will be subject to a second “meaningful vote” early next month.
Mrs May held phone discussions with key EU leaders over the course of the day, and will now step up efforts to win significant concessions from Brussels over the next two weeks.
The Prime Minister faces a tough time, however, as EU and Irish leaders have repeatedly insisted they will not reopen the Agreement.
Speaking ahead of the Commons showdown, Ireland’s European affairs minister, Helen McEntee, cautioned: “There can be no change to the backstop.”
Yorkshire MP Yvette Cooper said the Prime Minister was “running out of time” to make progress on Brexit as she made an unsuccessful plea to the Commons to back her amendment.
The Normanton, Pontefract and Castleford MP’s call for an extension of Article 50 to keep the UK in the EU until the end of the year in order to reach a deal was defeated by 321 votes to 298, majority 23.
Speaking earlier in the Commons debate, the Labour MP said she did not think the Prime Minister was “instilling confidence” or that she had a plan, adding: “I am really worried that the delay and the drift and the chasing of unicorns mean we could now end up with no deal by accident.”
Unless the Prime Minister changes direction and changes approach, she said: “I fear we will reach the brink.”
She added: “We can’t be cowardly about this. The Prime Minister is running out of time. Too few dare say it but everyone knows it. So before it is too late we have to be honest about this.
“And I urge people to support this amendment, to give the chance to discuss this Bill because if we can’t be honest about this at such a historic time I don’t know what politics is for.”
But Tory Andrew Percy, representing Brigg and Goole, criticised the amendment. He said: “Millions of people across the North of England voted in huge numbers to leave the European Union.
"Many of them then went out and elected their Labour MPs on a solemn commitment to make good on the referendum.
“Is it not the case that if any member of parliament representing a northern leave constituency votes for amendment B this evening they will be voting to dishonour the referendum result?”