However, it was received with caution by British MPs unsure of the legal ramifications, leaving Britain's exit from the European Union (EU) still precariously poised ahead of the March 29 Brexit deadline.
In a last-minute dash to the European Parliament in Strasbourg, France, on Monday night, May emerged alongside European Commission President Jean Claude-Juncker to declare that the UK and EU have agreed "legally binding" changes to the controversial Irish backstop clause to ensure any such arrangement would not be permanent.
The move is aimed at addressing the concerns of hard-Brexiteers in her own Conservative Party and the Northern Irish Democratic Unionist Party (DUP), which provides her government with its majority in the House of Commons.
"MPs were clear that legal changes were needed to the backstop. Today we have secured legal changes. Now is the time to come together to back this improved Brexit deal," May said at a joint press conference with Juncker.
The EU said it had made significant concessions as two additional documents were agreed to back up the withdrawal agreement struck in December last year a joint legally binding instrument which the UK could use to start a "formal dispute" against the EU if it tried to keep the UK tied into the backstop indefinitely and a joint statement committing both sides to find an alternative to the backstop by the end of the Brexit transition period of December 2020.
"In politics, sometimes you get a second chance. There will be no third chance it is this deal or Brexit might not happen at all," Juncker said, issuing a stark warning to Britain's MPs over the importance of the parliamentary vote in the UK on Tuesday.
Ireland's Indian-origin Prime Minister, Leo Varadkar, also stressed that the new agreements were an "unambiguous statement" of both sides' "good faith and intentions" even if they did not "undermine" the principle of the backstop or how it might come into force.
The Irish backstop, an insurance policy designed to maintain an open border on the island of Ireland between UK territory Northern Ireland and EU member-state the Republic of Ireland has been the biggest sticking point for many MPs in the UK who voted against the withdrawal agreement tabled by Theresa May in January, by a massive margin of 230 votes.
The historic bruising for May had led her to seek changes to the agreement in order to table a second meaningful vote before Parliament that would stand a better chance of being voted through.
Brexiteers from within her own party and the DUP have refused to comment if they feel the changes she has secured will be enough for them to vote in favour of the deal before they take full legal advice on the changes.
The Opposition Labour Party, meanwhile, has declared that the British PM has secured nothing new.
"This deal isn't the right deal. The PM is giving us basically the same deal," said Labour's shadow Brexit secretary Kier Starmer, who is calling for a "pause" and more time to consider the next steps.
The legal arguments around the new changes will be the focal point ahead of the vote scheduled for around 1900 GMT, with UK Attorney General Geoffrey Cox set to update his advice on the deal and whether it will stand up to legal scrutiny and be enough to convince sceptical MPs to vote in favour of May's deal.
If, however, the deal fails to pass through yet again, Theresa May's previously set timetable is set to kick in with MPs given a vote by Wednesday on leaving the EU without any agreement in place, which is expected to be defeated because there is very little support for such a no-deal Brexit option.
All eyes will then be on another Commons vote, expected by Thursday, in favour of delaying the March 29 deadline.
The March-end deadline had kicked in when the UK triggered Article 50 of the Lisbon Treaty following a referendum in favour of Brexit by nearly 52 per cent to 48 per cent or 17.4 million votes to leave the EU as opposed to 16.1 million to remain in the 28-member economic bloc in June 2016.