TSSA and the RMT called off action after talks with mediator Acas.
“We have now agreed a process where all our serious concerns over safety and job losses will be seriously addressed through the normal channels. We are obviously pleased that we have agreed this process,” a TSSA spokesman said.
The dispute centres on plans to close all 260 Tube ticket offices and cut 960 jobs.
Transport for London says it would save £50 million a year.
RMT General Secretary Bob Crow said: “After two days of intensive and detailed discussions through the offices of Acas we have now received proposals that halt the implementation of the job cuts set out in the HR1 document which gives us the opportunity to discuss all of the issues away from the pressure cooker.
“We now have a golden opportunity to look again in detail at all of the concerns we have raised about the impact of the cuts on our members and the services that they provide to Londoners. That is exactly what we have been calling for throughout this dispute.
“RMT is happy to discuss any issues with LU through the machinery of negotiation and we are glad that we have now got back to where we should have been right at the start of this process.
“It is unfortunate that we were forced and provoked into a dispute that we never wanted and we are now in a position to move on with the clear understanding that our action is suspended but if there is any further attempt to impose change from above the action will go back on.”
The strike was due to start at 21:00 GMT.
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