It came after the prime minister and Ursula von der Leyen had what the European Commission president described as a “constructive and useful” phone conversation on Sunday.
However, Johnson struck a markedly more downbeat tone, warning that the two sides are “still very far apart” on key issues.
Negotiations are now going down to the wire, with the UK due to default to World Trade Organisation terms for trade with the EU from January 1 – widely predicted to be the most damaging outcome – if agreement cannot be found.
However, there is increasing but cautious optimism that agreement may be possible with both sides deciding to continue negotiations without setting a new deadline, despite the PM claiming no deal was “very, very likely” on Friday.
In a joint statement, Johnson and von der Leyen said: “We had a useful phone call this morning. We discussed the major unresolved topics.
“Our negotiating teams have been working day and night over recent days.
“And despite the exhaustion after almost a year of negotiations, despite the fact that deadlines have been missed over and over we think it is responsible at this point to go the extra mile.
“We have accordingly mandated our negotiators to continue the talks and to see whether an agreement can even at this late stage be reached.”
Later, Johnson warned that there are still “serious difficulties” in the talks.
He told broadcasters: "As things stand, and this is basically what Ursula and I agreed, I’m afraid we’re still very far apart on some key things.
“But where there’s life, there’s hope.
“We’re going to keep talking to see what we can do, the UK certainly won’t be walking away from the talks, I think people expect us to go the extra mile.”
Talks between UK chief negotiator Lord Frost and EU counterpart Michel Barnier went on until 9.30pm on Friday, with their technical teams continuing to midnight.
The pair will now continue their negotiations in Brussels with talks stuck on the key issues of fishing rights and the extent to which the UK and EU would maintain similar standards.
Before the announcement, foreign secretary Dominic Raab said there is “a long way to go” to find a resolution on fisheries and so-called level playing field “ratchet” clause, which could see either side punished with trade tariffs if they do not bring their standards in areas like the environment up to the level of the other.
But Raab told BBC One’s Andrew Marr Show the bar that would need to be crossed to continue talking is “quite high”, suggesting that some movement must have been achieved for Johnson and von der Leyen to reach that conclusion.
“We would need at a political level a commitment to move on those two key issues,” he said.
“When you look at what I said at the start of the week, it is pretty similar to what I’m saying now, which is never say never because EU negotiations can often drag and drift.
“But, actually, we do need finality and therefore we need at the political level of Ursula von der Leyen, the president of the European Commission, that there is clarity the EU will move on those two key issues. If we get that then there are still talks to be processed.”
The row over what rights EU trawlers would have to access UK waters dramatically escalated over the weekend, with Royal Navy vessels on stand-by to patrol the seas around Britain if there is no deal in a move branded “irresponsible” by senior Tory MP Tobias Ellwood.
French president Emmanuel Macron is said to have threatened to veto a UK-EU trade deal after expressing dissatisfaction at the new quota terms being thrashed out for his fishermen.
Raab told Marr it is “crazy” of the French to make “aggressive demands” when a no deal scenario would leave their boats with “zero assured access” to UK fishing grounds.
He said the UK is “going to exercise full control in the way I think people would expect over our fisheries” but said any enforcement would be “proportionate”.
But Spanish foreign minister Arancha Gonzalez Laya said she does not believe the proposition of deploying gunboats is “serious”.
She told the Sophy Ridge programme: “I think this is all for the gallery, I don’t think this is serious – and, by the way, I don’t think this is needed.
“I think what would be more responsible is to sit down and agree what kind of relationship does the UK want with the European Union on fishing, again understanding that on this, like on the rest (of the issues), there are things for the UK to win, things for the EU to win – we just have to find this middle point.”
Responding to the news that negotiations were extended, a spokesperson for the Labour Party said: “The Conservatives promised the British people that they had an oven-ready deal and that they would get Brexit done.
“The government needs to deliver on that promise, get us the deal and allow us to move on as a country.”
The Confederation of British Industry (CBI)’s director-general Tony Danker said: “The news that talks will continue gives hope.
“A deal is both essential and possible.
“It is the only way to build upon the extensive support for the economy given by all governments during the pandemic.
“Without it, that progress is undermined.”