May said at a news conference overnight on Friday that Brussels approved pushing the deadline to May 22 if British MPs next week back a withdrawal agreement that they rejected twice before.
If the accord is rejected, Britain will have until April 12 to offer a new plan or choose to quit without a treaty.
The UK prime minister insisted she could secure a deal next week, saying British MPs have a “clear choice”.
“What this decision tonight does is show the clear choice that is open to MPs,” she said. “I think the choice is clear for people.”
Until April 12, said summit chair Donald Tusk, “all options will remain open and the cliff-edge date will be delayed”.
“The UK government will still have a choice between a deal, no deal, a long extension or revoking Article 50 (the withdrawal notice),” he told a news conference.
Despite an extraordinary joint warning from business and trade union leaders that the economic disruptions of a no-deal Brexit would present a “national emergency”, May had refused to rule out walking away.
“What the decision today underlines is the importance of the House of Commons passing a Brexit deal next week,” she said.
“Tomorrow morning, I am returning to the UK and working hard on building support to get the deal through… I hope we can all agree we are now at the moment of decision.”
Her comment comes just days after the speaker of the House of Commons blocked her proposal to hold a third vote on the deal, citing 400-year-old precedent that says the same proposition cannot be put to MPs again and again.
“There is a fairly even split between those who believe that if her deal collapses next week for the final time, May will simply throw caution to the wind and announce that the UK is leaving the EU with no deal at all, and others who think it is so inconceivable that she must have some sort of alternative arrangements,” said Al Jazeera’s Laurence Lee from Brussels.
“The EU leaders asked her straight out at this meeting if she had a Plan B, as usual, she refused to answer.”
There is also a growing impatience that the efforts to avoid a messy Brexit are bogging the EU down when it has other priorities, from a weakening economy to rising nationalism.
French President Emmanuel Macron has led the charge, calling for Britain to put up or shut up on Brexit.
“It is up to the British to sort out their own internal contradictions. As for us, we don’t have any.
“The responsibility now lies with the British, and I think that’s the big achievement of the day.”
Al Jazeera and news agencies