The strike had been organised due to overwhelming support for a strike after a "comprehensive breakdown in industrial relations".
The union said the strike had been suspended after progress was made in talks with London Underground bosses.
RMT general secretary Mick Cash said: "The hard work of RMT reps, backed up by the solidarity and determination of RMT members, has enabled us to make enough progress in talks over the issues at the heart of this dispute to allow this week’s action to be suspended and to allow for those talks to continue.
"However, the dispute is not over yet and RMT pays tribute to the Piccadilly line members whose resolve has been instrumental in pushing the negotiations forwards."
The Piccadilly is the fourth busiest line on the London Underground network transporting an average of 600,000 passengers a day and services London’s Heathrow airport – the busiest airport in Europe.
Pat Hansberry, operations director for London Underground, said: "We welcome the RMT and ASLEF’s decision to withdraw the strike action that was planned for this week. Both parties have engaged in constructive talks which have enabled us to reach an agreement on the way forward to resolve this dispute."