Marseille, aiming for a European place, continued to push forward and had another chance midway through the first half through Benoit Cheyrou, but his finish on goal was saved.
Rennes, who are fighting against relegation from the French top flight, drove upfield just two minutes later, however, and Romain Alessandrini was able to get in a shot, but his strike was blocked.
With five minutes left to play of the first half, Andre Ayew smashed in a shot on target, but Costil was again able to thwart the opposition.
Marseille picked up from where they left off at the start of the second period, with Cheyrou getting a shot away straight after the restart.
The stunned hosts desperately tried to claw a goal back in the final 15 minutes, but it was to no avail.
The result means that Rennes are now six points above the drop zone, while Marseille are four points off fourth place, having played a game more than the teams around them.