Members of Aslef and the Rail, Maritime and Transport union (RMT) were due to take industrial action throughout the week in a long-running dispute over pay, pensions and conditions.
Both unions said progress had been made in talks at the conciliation service Acas.
Finn Brennan, Aslef’s organiser on the Underground, said: “After a week of intense negotiations, we have made real progress in making sure our members’ working conditions and pensions are protected from the impact of the Tory Government cuts to Transport for London (TfL) funding.
“There will be no changes to pension benefits before the next general election and any future changes to working conditions and agreements will only be made by negotiation.”
RMT general secretary Mick Lynch said: “There has been significant progress made by our negotiating team in Acas talks with TfL.
“However, this is not the end of the dispute nor is it a victory for the union as yet.
“Our members were prepared to engage in significant disruptive industrial action and I commend their resolve.
“RMT’s strike mandate remains live until October and we are prepared to use it if necessary.
“We will continue to negotiate in good faith as we always have done with TfL and it was only the steadfast commitment of our members in being prepared to take sustained strike action that has forced the employer to make significant concessions.
“Our campaign to defend jobs, conditions and our members’ pensions will continue in the coming weeks and months.”
The Mayor of London Sadiq Khan welcomed the news, saying: “Despite the onerous funding deal conditions imposed by the Government, we have managed to avoid industrial action.
“Negotiation is always the best way forward and this shows what we can achieve by working with trade unions.
“I’ve been in close contact with the TfL commissioner throughout this week, and I’ll keep working with our TfL unions and staff to deliver the best transport system in the world for Londoners.”