A union official said progress had been made during talks to resolve a dispute over planned job cuts, so there was no need to walk out.
The planned work stoppage would have involved border guards, including immigration and passport workers.
With the Olympics opening ceremonies just two days away, a Thursday strike would have created huge bottlenecks at Heathrow just as thousands of tourists descend on the British capital.
Heathrow officials say Thursday is expected to be the airport's busiest day ever.
Britain's Home Office said it welcomed the decision to cancel the job action.