George Ford was the key man yet again as his five penalties secured an impressive away victory at the Stade Chaban Delmas.
Ford kicked Bath into a 6-0 lead with penalties after nine and 27 minutes before Bordeaux hit back with three points from scrum-half Manu Saubusse when second row Will Spencer was yellow carded seven minutes before the break.
A third penalty from Ford handed Bath a 9-3 advantage on the stroke of half-time and the young fly-half made it 12-3 with another 10 minutes after the interval.
Paul James became the second Bath player to see yellow after 64 minutes and Saubusse cut the gap with his second strike with seven minutes left to play.
But Bath held firm and Ford secured the win with his fifth success moments before the final whistle.