Montpellier (13) 32
Tries: Kuzbik, Doumayrou Cons: Lagarde 2 Pens: Lagarde 6
Exeter (17) 30
Tries: Arscott, Jess 2 Cons: Davis 3 Pens: Davis 2
Exeter just missed out on a place in the last eight of the European Challenge Cup as Montpellier came from behind to take the spoils.
Luke Arscott and Matt Jess tries put Exeter ahead, before Sebastien Kuzbik pulled a try back for Montpellier.
Jess scored his second try shortly after the break, but six penalties from Raphael Lagarde kept Montpellier within touching distance.
Geoffrey Doumayrou then scored the winning try to put Montpellier through.
In near perfect conditions at Stade Yves-du-Manoir, Exeter dominated the opening exchanges, their first try coming after seven minutes as centre Sireli Naqelevuki stormed through to release Arscott.
Naqelevuki then provided the assist for Exeter's second, again using his strength to shrug off several tackles before releasing Jess, who made no mistake with the finish.
Kuzbik's try kept the home side in touch and the half ended with Exeter leading by 17-13.
When Jess went over for his second, Exeter looked to be in control, but Lagarde's ever-reliable boot kept the contest open.
A yellow card for replacement centre Brian Rennie after 67 minutes handed Montpellier an advantage they wasted little time in using, Doumayrou scoring the converted try that gave them the lead for the first time.
Exeter threw everything at their hosts as they looked for a way back, but all their effort in the dying minutes came to nothing as Ryan Davis pushed an attempted drop-goal wide.
Exeter head coach Rob Baxter told BBC Radio Devon:
"The boys did a lot of really good stuff, battled away and scored some good tries, but at the same time there were enough errors to never really let us get away from Montpellier.
"It was a tough game. There's been a fair bit of blood flying around and in a tough game sometimes little mistakes happen, and that's what happened out there.
"We haven't been in many games where you try and win it with a drop-goal at the end and those are the things that we have to learn as a team."
Montpellier: Alcalde, Pratmarty, Tuilevuka, Rees, Kuzbik, Lagarde, Paillaugue, Leleimalefaga, Caudullo, Kervarec, Uva, Demarco, Bost, Wasserman, Giraud.
Replacements: Q. Doumayrou for Pratmarty (56), Nicot for Rees (71), Tomas for Paillaugue (70), Shvelidze for Kervarec (51), Macurdy for Demarco (49).
Not Used: Rofes, Petit, Chkhaidze.
Exeter: Arscott, Higgins, Dollman, Naqelevuki, Jess, Davis, Poluleuligaga, Moon, Clark, Budgen, Hayes, Hanks, Johnson, Scaysbrook, Baxter.
Replacements: Thomas for Poluleuligaga (47), Sturgess for Moon (57), Whitehead for Clark (59), Tui for Budgen (56), Phillips for Johnson (69), Slade for Scaysbrook (59).
Not Used: Steenson
Sin Bin: Rennie (67).
Referee: Dudley Phillips (RFU)