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After a meeting that stretched through the afternoon and over dinner on Thursday, the bloc said Britain could postpone its March 29 departure to May 22 - if the UK Parliament approves Prime Minister Theresa May's divorce deal with the bloc next week.
If the twice-rejected deal is thrown out again, the bloc says Britain has until April 12 to "indicate a way forward."
"Now it is finally up to the British political system," French President Emmanuel Macron said, adding that any final decision must come ahead of the European Parliament election from May 23-26.
May - who has spent almost three years telling Britons they will leave the EU on March 29 this year - put a positive spin on the delay.
She said the EU decision underlines "the importance of the House of Commons passing a Brexit deal next week so that we can bring an end to the uncertainty and leave in a smooth and orderly manner."
The late-night offer eased some of the deep uncertainty among leaders at an EU summit in Brussels, which was exceeded only by the high anxiety being felt by politicians, businesses and citizens in Britain. The British military has even set up a command post in a bunker under the defence ministry in London to help coordinate "no-deal" planning.
The House of Commons is split, both among and within its political parties, over whether and how to leave the EU. It has twice rejected the deal May brokered with the bloc's leaders late last year.
This week, May finally acknowledged the Brexit gridlock and asked the EU to delay Britain's departure until June 30, to create time to win parliamentary approval for her deal in a third attempt and then pass the legislation necessary for a smooth departure.
But opposition to May's the agreement among British politicians appeared to be hardening, rather than softening, after she blamed Parliament for the Brexit impasse.
In a televised address on Wednesday night, May accused lawmakers of "infighting," ''political games" and "arcane procedural rows," but acknowledged no personal error in creating the deadlock.
An MP from May's Conservative Party called the speech "toxic." Anna Soubry, of the breakaway Independent Group, described it as the "most dishonest and divisive statement from any Prime Minister."
May struck conciliatory note at a late-night Brussels news conference, saying "I know MPs on all sides of the debate have passionate views, and I respect those different positions."
"Last night I expressed my frustration. I know that MPs are frustrated too. They have difficult jobs to do," she added.
But May also refused to change course, calling on MPs to back her agreement and refusing to rule out a no-deal exit if they did not back her.
May said that if the deal falls, by April 12, "we would either leave with no deal, or put forward an alternative plan" that involved participating in EU Parliament elections.
"I believe strongly that it would be wrong to ask people in the UK to participate in these elections three years after voting to leave the EU," she said.
Businesses and economists say a no-deal Brexit would cause huge disruptions and billions in costs to the economies of both Britain and the EU.
Underscoring the sense of dread gripping the nation, one of Britain's biggest business lobbies and a major trade union federation said in a rare joint appeal that the "country is facing a national emergency."
The Confederation of British Industry and the Trades Union Congress warned May that if Britain crashes out of the EU, "the shock to our economy would be felt by generations to come."
Britain's military said the command post under the ministry of defence was set up as a continuation of earlier planning operations, called Operation Redfold, designed to minimise disruption in the event of a no-deal Brexit.
The ministry said in a statement it had 3500 troops on standby to help with any disruptions if the government asks for assistance.
Worry about a chaotic departure has been rising among EU leaders, who fear May no longer has the clout in Parliament to get her way.
"Nobody wants no-deal here," Irish Prime Minister Leo Varadkar told reporters.
German Chancellor Angela Merkel vowed to work "until the last hour" to try to ensure that Britain doesn't leave without a deal, even though her government has enacted emergency measures to deal with such a scenario.
May plans to make a third attempt to get her deal through Parliament next week. But many pro-Brexit legislators still oppose it, saying it does not deliver the clean break they long for.
And Pro-EU lawmakers will try to derail May and wrest away control of the Brexit process to steer Britain toward a close relationship with the bloc.
It's a struggle that has been going on for almost three years and has brought the UK to within eight days of a chaotic Brexit.
EU leaders are watching with disbelief, horror - and, for some, sympathy.
"I never got frustrated with Theresa May. I have the highest respect for her," said Dutch Prime Minister Mark Rutte. "Her tenacity is enormous. But she is working in an extremely difficult situation.
"It's not her mistake that we are where we are - it's because too many people have so far played party politics on this issue."