Exeter reached the Champions Cup quarter-finals by scoring more tries than 14-man Montpellier in a thriller that finished 33-33 after extra time.
Joe Simmonds converted Jack Yeandle's try in the 100th minute to level the scores, with Chiefs having scored five tries to Montpellier's four.
The visitors had Zach Mercer sent off in the second half but Paolo Garbisi's penalty took the game into extra time.
Louis Carbonel's try put Montpellier ahead before Exeter's late heroics.
Chiefs, the 2020 champions, will host South African side Stormers at Sandy Park next weekend for a place in the last four.
The last-16 tie had started dramatically, with Montpellier losing hooker Brandon Paenga-Amosa to a leg injury in the third minute but his replacement Curtis Langdon scored the game's first try in the left corner soon after.
Exeter were soon further behind when Vincent Rattez crossed for Montpellier, but the hosts clawed their way back when prop Scott Sio went over for a converted try.
Chiefs took the lead when Joe Simmonds converted his brother Sam's try, and Exeter were in control until Garbisi landed a penalty to give the visitors a 15-14 advantage at half-time.
Montpellier seemed to have gone further ahead with a try from former Bath number eight Mercer early in the second half until it was ruled out for an infringement in the build-up.
Mercer, the Top 14 player of last season, was then dismissed for a shoulder-to-head tackle on Christ Tshiunza soon after, but Garbisi's second penalty extended Montpellier's lead.
However, Exeter hit back with tries from Tom Wyatt and Josh Iosefa-Scott putting them 26-18 in front, while Montpellier were frustrated by Thomas Darmon's score being disallowed for a knock-on.
Lenni Nouchi then went over near the corner flag, but Garbisi missed the difficult conversion to leave the game at 26-23 as the clock ticked past 80 minutes. But the Montpellier fly-half made amends with a 45-metre penalty that took the game into extra time.
Exeter were held up over the line late in the first added period, while Carbonel's penalty attempt from his own half went just under the bar, but the Montpellier replacement soon sliced through a gap for a converted try to make it 26-33.
But Exeter were not done, and Yeandle crossed just to the right of the posts with Joe Simmonds slotting the crucial kick to seal Chiefs' place in the last eight.
Line-ups
Exeter: Wyatt; Nowell (capt), Slade, O'Brien, Woodburn; J Simmonds, Becconsall; Sio, Frost, Street, Gray, Jenkins, Kirsten, Tshiunza, S Simmonds.
Replacements: Yeandle, Hepburn, Iosefa-Scott, Davis, Ewers, Cairns, Skinner, Hodge
Montpellier: Bouthier (capt); Lam, Darmon, Serfontein, Rattez; Garbisi, Reinach; Forletta, Paenga-Amosa, Thomas, Stooke, Duguid, Van Rensburg, Dakuwaqa, Mercer.
Replacements: Langdon, Lamositele, Haouas, Nouchi, Coly, Carbonel, Bridge, Tauleigne.
Referee: Andrew Brace (Wal)