The suspension follows successful talks at the conciliation service Acas, according to RMT.
The union said "all objectives in the Piccadilly line dispute" had been achieved, while the union is "committed to further talks" aimed at a long-term agreement on the Hammersmith and City line.
RMT general secretary Mick Cash said:
Talks with London Underground have proved to be positive and focussed and as a result we have been able to make enough significant progress to allow us to suspend the strike action in both disputes that was scheduled for tomorrow evening.
There is no question that the advances we have made have been down to the strength and resilience of our members in the workplace backed up by a determined and professional negotiating team. The solidarity that led to the resounding ballots for action has been absolutely decisive.