'Additional six months for UK to find best possible solution'
May had asked for a delay only until June 30.
Tusk said in a tweet that the British leader had agreed to the longer "flexible" extension, which means Britain can leave before October if it ratifies a withdrawal deal with the EU.
"This means additional six months for the UK to find the best possible solution," Tusk wrote.
EU leaders held the emergency summit two days before Brexit
Just two days before Britain was due to leave the EU, its leaders spent a long dinner meeting wrangling over whether to save Britain from a precipitous and potentially calamitous Brexit or to give the foot-dragging departing nation a shove over the edge.
May pleaded with them at an emergency summit to delay Britain's exit, due on Friday, for a couple more months.
Nothing should compromise European projects in the coming months: Macron
Some were sympathetic, but French President Emmanuel Macron struck a warning note shortly before the European leaders met.
"Nothing is decided," Macron said as he arrived at the summit, insisting on "clarity" from May about what Britain wants.
"What's indispensable is that nothing should compromise the European project in the months to come," he said.
May's plea for Brexit-delay was rejected three times before.
June-30 deadline was enough time to ratify Brexit deal: May
May said a June 30 deadline was enough time for Britain's Parliament to ratify a Brexit deal and pass the legislation. But British-lawmakers have rejected her divorce deal three times and attempts to forge a compromise with her political opponents have yet to bear fruit.
The summit also had signs of warmth and humor
May spoke to the 27 EU leaders for just over an hour, before they met for dinner without her to decide Britain's fate.
There were signs of warmth and humor in this summit.
May and German Chancellor Angela Merkel were filmed laughing over a tablet bearing an image showing the two of them speaking to their respective Parliaments yesterday wearing similar blue jackets.
May hadn't offered "sufficient guarantees" to justify long-extension: Macron's office
Many leaders said that they were inclined to grant a Brexit delay, though Macron had reservations after hearing May speak.
An official in the French president's office said the British leader hadn't offered "sufficient guarantees" to justify a long extension.
Meanwhile, other leaders suggested that a longer delay would likely be needed, given the depth of Britain's political disarray.
Was hopeful withdrawal-agreement would be ratified by May 22: Theresa
After arriving in Brussels, May said, "What is important is that any extension enables us to leave at the point at which we ratify the withdrawal agreement."
However, she added that she was hopeful it could be as soon as May 22, a key date, since that would avoid the need for Britain to participate in elections for the European Parliament.
UK lawmakers rejected the Brexit deal three times
Several months have passed since May and the EU struck a deal laying out the terms of Britain's departure and the outline of future relations. All that was needed was ratification by the British and European Parliaments but it faced rejections three times by UK-lawmakers.
No-deal Brexit would disrupt trade and travel: Economist
As Britain's departure date of March-29 approached with no resolution in sight, the EU gave Britain until Friday to approve a withdrawal plan, change course and seek a further delay to Brexit, or crash out of the EU with no deal to cushion the shock.
Economists and business leaders have warned that a no-deal Brexit would lead to huge disruptions in trade and travel.
May promised to step down once Brexit is delivered
A no-deal Brexit would cause gridlock at British ports and possible shortages of goods. It would hurt EU nations as well as, Britain.
May's future, meanwhile, is uncertain. She has previously said that "as prime minister" she couldn't agree to let Britain stay in the EU beyond June 30, and she has also promised to step down once Brexit is delivered.
May survived a no-confidence vote in December
Many Conservative Party lawmakers would like May to quit now and let a new leader take charge of the next stage of Brexit. But they can't force her out until the end of the year after she survived a no-confidence vote in December.
Labour favors a softer Brexit than the government has proposed
Every British initiative to get a deal has floundered so far.
Several days of talks between May's Conservative government and the opposition Labour-Party aimed at finding a compromise failed to produce a breakthrough.
Labour favors a softer Brexit than the government has proposed, and wants to retain a close economic relationship with the bloc.
The two sides said they would resume their discussions today.