The union said it had carried out a "long and hard fought campaign", which had resulted in "significant progress in the talks".
"[That] has enabled us to now suspend all industrial action in the station staffing dispute," a spokesman said on Friday afternoon.
"The breakthrough has been secured through the solidarity of the workforce and their willingness to take industrial action to defend jobs and safety."
The strike, which would have run from 9pm on Saturday 6 February and for the whole day on Sunday and Monday, would not have shut off the entire network, but would have meant a number of Tube stations were shut.
“I welcome the RMT’s decision to call off the strike action and end this dispute," said Steve Griffiths, London Underground's chief operating officer.
"Our station modernisation programme now continues, with our changes meaning a more personal service for customers, and record numbers of staff in public areas visible and available to help customers where they need it most.
“As we’ve always promised, we’re delivering these improvements with no compulsory redundancies, no-one being asked to work more hours or lose any pay, as well as continued focus on work-life balance within rosters. In addition, around 900 staff are either being promoted or securing permanent roles with London Underground, having previously been employed on fixed term contracts.”
However, a further seven sets of industrial action are set to continue as the union fights over safety issues on behalf of its maintenance staff members.