Bordeaux book Champions Cup quarter-final with victory over Ulster
They face a home tie against another Irish province Munster next weekend while Toulouse can make it four French teams if they can turn over Sale later on Sunday.
A sunny afternoon at the Stade Chaban Delmas inspired a vintage game of free-flowing rugby from both sides with Bordeaux's six tries answered by five from the visitors.
"We started well, but finished (the match) badly," said Bordeaux coach Yannick Bru.
"In six days' time, the intensity will be greater because Munster are superior to this Ulster team who were true to their values, moving the ball, very physical and present for 80 minutes.
"We made a few mistakes that gave us a few scares."
There was no Louis Bielle-Biarrey in the Bordeaux line-up but it took just seven minutes for France's other great wing, Damian Penaud to make his presence felt.
A drifting run from inside the Bordeaux half that brought him in off the right wing, ghosting past a host of defenders to dot down under the posts, was his 11th try this season, equalling Saracens' Chris Ashton's competition record set in 2013-2014.
Former Ireland fly-half Joey Carbery, starting ahead of Matthieu Jalibert, converted and was on hand to add the extras 10 minutes later when prop Ben Tameifuna crashed over from close range for their second try.
Ulster fightback
Moments later the former Australia and Tonga lock Adam Coleman stretched out a long right arm to touch down. With Carbery's touchline conversion Bordeaux were 21-0 up and on track for a runaway win.
Ulster, the 1999 champions, hit back quickly, however, with Jack Murphy and Jacob Stockdale breaking into the 22 before Stuart McCloskey's quick penalty caught the hosts on the back foot and prop Tom O'Toole powered over.
Ulster then showed real confidence by eschewing a kickable penalty in front of the posts to go for the scrum. After several phases it resulted in David McCall squeezing through for a second try.
John Cooney converted both to bring it back to 21-14 but Bordeaux had the last say in the half, Carbery starting a glorious flowing movement that went through six pairs of hands before Maxime Lucu fed Romain Buros to go in at the corner.
Carbery converted to put Bordeaux 28-14 up at the break but it was to be his last significant action. Injured at the start of the second half, he was replaced by France international fly-half Jalibert.
The home side was again quick out of the blocks. Maxime Lucu popped over a penalty before hooker Maxime Lamothe was driven over after a line-out close to the Ulster line.
At 36-14, it again looked as though Bordeaux were home and dry but once again, Ulster bounced back with Nick Timoney and Zac Ward both touching down although replacement Nathan Doak missed both conversions.
Bordeaux always seemed to have another gear, though, and when Rohan Janse van Rensburg went over with nine minutes remaining, the game was up for Ulster, although Ward snatched his second just before the end.
"I'm extremely proud of the lads and the way they stood their ground and stayed in the fight," said Ulster coach Richie Murphy. "We played some really good rugby at times."