Both the UK and EU had set a deadline of Sunday to decide whether the tortuous negotiations should be abandoned – but have decided to step back from the brink.
However, there is no sign of a decisive breakthrough on the flashpoints of fair competition rules and fishing rights, leaving open the prospect of a no-deal outcome in just 18 days’ time.
Unlike last week – after failed face-to-face talks in Brussels – no fresh deadline was set for progress being made.
In a joint statement, the prime minister and Ms 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.”
Neither side has been willing to walk away from the negotiating table – risking the blame for the huge disruption to borders and to economies from no agreed trading terms.
In London, the prime minister briefed his Cabinet on the decision before the joint statement was released.
In Berlin, the German Chancellor Angela Merkel welcomed the decision, saying: “Every opportunity to reach a deal is highly welcome.”
Despite the shared relief that a no-deal has been averted – for now – the UK is still warning that the EU will have to move “a long way” for the talks to succeed now.
Dominic Raab, the Foreign Secretary, attacked Brussels’ “outlandish” demands which “don't get more reasonable by repetition”.
He condemned the threat of a “torpedo of tariffs”, reflecting the UK’s anger that the EU wants to impose unilateral punishments for perceived ‘level playing field’ rule breaches.
Mr Raab said London wanted “to narrow the issues if any disputes arise in the future”, hinting it might accept the threat of tariffs restricted to a particular sector.
But he criticised “a nuclear-style type reaction”, of wider tariffs that “go up and we are back in the same old drama and soap opera every couple of years or even sooner than that, just because there is a particular issue around a particular sector”.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies