Glasgow were knocked out of the Champions Cup after Montpellier secured a bonus-point win over Exeter to claim the final last-16 spot.
Cobus Reinach's 70th-minute try confirmed Montpellier's progress to the next stage at Glasgow's expense, with the French side setting up a knockout tie against Harlequins.
Sam Simmonds impressed for 2020 winners Exeter, scoring a hat-trick.
The Chiefs will play Munster for a place in the quarter-finals.
The Irish province thrashed Wasps earlier on Sunday to claim third spot in Pool B.
For Montpellier, the result lifted them above Clermont into seventh place in Pool A and the Top 14 side can now look forward to a repeat of the 2016 Challenge Cup final against Premiership champions Harlequins.
Montpellier went into the game needing a result having been hammered 89-7 by Leinster in Dublin last week, but it was Exeter who got off to a flyer. After just three minutes, Simmonds burst around the front of a close-range line-out and rode three tackles to cross and mark his 100th Chiefs appearance in style.
That served only to spark the hosts into life, however, as Gabriel N'Gandebe demonstrated electric pace to surge through a gap in the Chiefs' defence to create Bouthier's try.
Five minutes later, lock Paul Willemse powered over following a driving Montpellier maul before Bouthier floated a brilliant pass out wide for Jan Serfontein to notch the Top 14 side's third try despite Stuart Hogg's attempted tackle on the line.
On the half-hour mark, Louis Foursans kicked over a penalty to stretch Montpellier's lead to 19 after Luke Cowan-Dickie was penalised for a no-arms tackle.
At that stage, Exeter appeared to be fighting a losing battle but with the home side having lost Bouthier to injury, the Chiefs produced a stirring response to haul themselves back into the game.
First, Simmonds notched his second try of the evening when he charged through the Montpellier defence close to the posts before centre Sean O'Brien intercepted prop Enzo Forletta's attempted pass out wide and ran in to score Exeter's third.
Joe Simmonds' conversion brought the Premiership side back to within five and they looked to have levelled on the cusp of half time when Hogg got over the line under pressure, only for the TMO to show the Scotland captain had knocked the ball on.
But Exeter did not let that decision deter them and they nudged ahead when Simmonds drove through two tackles to complete his hat-trick, his score the product of a sustained and patient spell of pressure.
Needing a bonus point, Montpellier pressed and went ahead when Foursans slotted over a straightforward penalty, but the telling moment arrived with 10 minutes remaining.
Reinach intercepted a stray Sam Maunder pass, ran virtually the length of the pitch and secured the hosts' bonus point, condemning Glasgow to the Challenge Cup.
Exeter director of rugby Rob Baxter, speaking to BT Sport: "We got it a little bit wrong at the start, but the middle third of the game was very good for us. We pulled things back and got in the ascendancy, but just couldn't see things off.
"We didn't make the most of our opportunities in the end and Montpellier capitalised on it. There was a lot more there for us than we took today.
"[On facing Munster] It'll be brilliant for our supporters. Munster is a great place to go and play rugby. We're looking forward to that, it'll be a nice experience for us."
Line-ups
Montpellier: Bouthier; N'gandebe, Reilhac, Serfontein, Rattez; Foursans, Aprasidze; Forletta, Paenga-Amosa, Haouas, Chalureau, Willemse, Van Rensburg, Camara (c), Mercer.
Replacements: Guirado, Rodgers, Thomas, Becognee, Reinach, Garbisi, Darmon, Verhaeghe.
Exeter: Hogg; Nowell, Slade, O'Brien, O'Flaherty; J Simmonds, J Maunder; Moon, Cowan-Dickie (c), Nixon, S Skinner, Lonsdale, Ewers, Kirsten, S Simmonds.
Replacements: Innard, Kenny, Schickerling, Tshiunza, Armand, S Maunder, H Skinner, Gilbert-Hendrickson.
Match officials
Referee: Frank Murphy (Ireland)
Assistant referees: Peter Martin & Daniel Carson (Ireland)
TMO: Olly Hodges (Ireland)