Bristol were knocked out of the Champions Cup after being overpowered by Bordeaux-Begles, who will host Racing 92 in the quarter-finals.
The Bears led for most of the first 40 after Henry Purdy crossed but Mathieu Jalibert scored 15 points from the tee to Callum Sheedy's nine to go ahead.
Bristol's hopes faded when France star Jalibert converted his own try.
Sheedy scored another penalty but Joseph Dweba and Nans Ducuing tries sealed victory.
The result means Exeter are the only Premiership side currently through to the quarter-finals, which will take place on the weekend of 9-11 April, but Sale could still join them if they beat Scarlets later on Sunday.
Bristol fade after strong start
Neither side had played in the Champions Cup knock-out stages before and it was Bristol who had the strongest start in the south of France, showing the slick attack that has helped them to the top of the Premiership.
The visitors - Challenge Cup winners last season - swung the ball left off a line-out, with returning full-back Charles Piutau creating enough space for wing Purdy to score.
That would be the only try of the first half, though, as indiscipline quickly took over the narrative. It was Jalibert who kicked the first penalty before Sheedy repaid the favour shortly after.
This pattern was repeated twice and, though Bordeaux gathered momentum towards the end of the first half, the France 10 had to settle for one last penalty to take his side ahead at the break.
The hosts struck first in the second half as France fly-half Jalibert scrambled across the tryline and, although referee Mike Adamson did not initially think a try had been scored, the television match official found evidence it had been.
Jalibert converted his own score to give Bordeaux an eight-point lead, before Bristol's Purdy crossed the tryline again but knocked the ball on as he fell to the floor.
The Bears continued to respond strongly and were compensated with another three points for Sheedy.
They were given further reason to hope after Jalibert missed a penalty and Kane Douglas was sin-binned in the 68th minute following one too many Bordeaux penalties at the breakdown.
But Bristol could not turn things around as Dweba sliced his way over against the grain and the score was given after a possible knock-on earlier in the move by Scott Higginbotham was investigated and dismissed.
Cyril Cazeaux joined Douglas in the sin-bin and it was Bordeaux who scored last as Ducuing intercepted a pass to claim a third and final try.
The result caps an impressive weekend for French sides in the competition as Top 14 teams make up five of the final eight.
Line-ups
Bordeaux-Begles: Buros; Moefana, Uberti, Lamerat, Lam; Jalibert, Lucu; Poirot, Maynadier, Cobilas, Cazeaux, Marais, Woki, Petti, Higginbotham
Replacements: Dweba, Paiva, Kaulashvili, Douglas, Roumat, Lesgourgues, Seuteni, Dulcuing.
Bristol: Piutau; Morahan, Radradra, Leiua, Purdy; Sheedy, Uren; Thomas, Byrne, Sinckler, Attwood, Joyce, Vui, Thomas, Hughes.
Replacements: Capon, Lahiff, Afoa, Holmes, Earl, Randall, Bedlow, O'Conor.