Sochaux took a surprise home lead in the 34th minute as Sanjin Prcic got in behind the defence to put his side in front.
Montpellier got back into the game straight after half time when Jonas Martin converted from the spot in the 48th minute to bring the score to 1-1.
The away side pushed for a normal-time victory in the second half, but had themselves to blame as they threw away a host of chances, while also being denied by some good defensive work from the hosts.
It looked as if the sides were headed for a penalty shootout as extra time had provided no real clear-cut chances, but Corchia popped up with a goal in the 115th minute, before Roy Contout extended the home side's advantage a few minutes later.
Montpellier did manage to pull one back in an exciting last few minutes through Benjamin Stambouli, but it was too little too late as the away side went crashing out of the tournament.