Exeter pushed Bordeaux-Begles close in France but a 34-27 defeat has left the Chiefs with a mountain to climb to progress in the European Champions Cup.
Dave Lewis, Byron McGuigan and Max Bodily all scored tries as a second-string Exeter rattled their French hosts.
But they fell agonisingly short in the end as Bordeaux claimed maximum points with a five-try performance.
Marco Tauleigne scored the winning try nine minutes from time after Adam Ashley-Cooper, Blair Connor, Louis-Benoit Madaule and Benat Auzqui had crossed earlier.
The result leaves Exeter with 11 points from five matches in Pool 2, five points adrift of leaders Ospreys, three behind Clermont Auvergne and level with Bordeaux.
All four sides still have a chance of getting through but Exeter, who host Ospreys in their final match, know they are now up against it.
Chiefs head coach Rob Baxter had made 13 changes to the side that beat Gloucester in the Aviva Premiership but Exeter still took the lead inside three minutes through the boot of Will Hooley.
Matt Jess and Elvis Taione both threatened for the visitors early on but it was Bordeaux who were dominating possession and they took the lead with two tries in quick succession.
Former Wallaby Ashley-Cooper picked up a loose ball from Julien Rey's break to cross after 13 minutes and then Connor followed just three minutes later from Pierre Bernard's pass.
Fly-half Bernard missed with both conversions to limit Bordeaux's lead to 10-3, but he was successful with his third shot at goal as he slotted over a 23rd-minute penalty.
Exeter were struggling for possession but hit back with a classic counter-attack. McGuigan ran back a loose kick before sending scrum-half Lewis racing over.
Hooley's conversion made it a three-point game but Bordeaux's response was immediate.
Yann Lesgourgues' burst along the left-hand touchline ended with captain Madaule driving over.
Yet Exeter again found a way back as Michele Campagnaro released McGuigan for a try that Hooley converted before adding a penalty to level the scores at 20-20 on the stroke of half-time.
Exeter were again forced to defend at the start of the second half as McGuigan prevented Metuisela Talebula touching down.
Bordeaux were not to be denied for longer though, as their forwards set up a driving maul from a lineout that ended with replacement hooker Auzqui touching down after 56 minutes for the bonus point and Lionel Beauxis converted.
Exeter refused to give up and three minutes later responded again when Bodily finished a sweeping attack.
Olly Woodburn found space before exchanging passes with Bodily who raced clear to score before the hour mark.
Hooley's conversion tied the scores at 27-27 but the fly-half was sin-binned in the 67th minute for infringing at a ruck as Bordeaux attacked.
Ashley-Cooper had a second try ruled out by the video official but replacement back-row Tauleigne broke clear a moment later for the decisive score.
Exeter had late chances and were helped when Bordeaux were reduced to 14 men for the final four minutes with a yellow card for prop Sebastien Taofifenua, but the visitors could not find a scoring pass to claim either a bonus point or the draw.