Striker Belkacem Zobiri shot past goalkeeper Laurent Pionnier from just inside the box to help the fourth-division side complete a second victory over a top-flight club, having already eliminated St Etienne on penalties in a previous round.
Montpellier, who beat Paris St Germain in the last 32, were awarded a penalty 12 minutes into extra time but Mbaye Niang, on loan from AC Milan, hit the post.
Cannes, a former Ligue 1 club where France greats Zinedine Zidane and Patrick Vieira made their professional debuts, had not reached the French Cup quarter-finals since 1992.
Lille edged a penalty shoot out with Caen 6-5 to progress to the last eight following an entertaining 3-3 draw while Angers SCO beat CA Bastia 4-2.
- Sports & Recreation
- Laurent Pionnier
- Patrick Vieira