The game's first real opportunity fell to Rennes when Anthony Mounier picked up the ball on the edge of the box, but he blazed his shot over the bar.
Montpellier could have gone ahead thanks to great play which found Remy Cabella inside the box, but his dink over the keeper was kept out by a last-ditch effort from Costil.
Cabella then missed a crucial chance just before the end of the half to put his side into the lead, as his half-volley flew just wide of the post.
The home side continued from where they had left off after half time and should have gone ahead but Victor Montano's shot brought a world-class save from Costil to keep the scores level.
Costil went on to have an outstanding second half, barring one error, allowing Rennes to maintain a clean sheet away from home and claim an important away draw.