George Ford celebrated his senior England call-up with another assured display with 17 points that included an impressive solo try.
Semesa Rokoduguni also grabbed a brace while Tom Biggs, Anthony Perenise, Jonathan Joseph, Dominic Day, and Kyle Eastmond also crossed in a 54-13 success to guarantee a quarter-final tie at the Recreation Ground.
Bath enjoyed a dream start when Biggs claimed the first of his team's eight tries after Anthony Watson's stunning 60-metre break with just a minute gone.
Ford's conversion made it 7-0 but Bordeaux hit back with a brace of penalties from scrum- half Manu Saubusse.
It took until half an hour for Bath to claim their next score but it was well worth the wait when Ford danced through from close range after 32 minutes.
Perenise then dived over out wide after Leroy Houston's long pass four minutes later, with Ford adding both sets of extras for a 21-6 lead at the break.
Joseph glided through from Ford's pass five minutes after the restart and Day powered through five minutes later when Ford's kick caused havoc at the back.
Ford added both conversions and did the same again when Rokoduguni flew down the right wing for a fine individual try with 58 minutes gone before finally missing the target when Eastmond was driven over just past the hour.
Bordeaux made the most of Ryan Caldwell's yellow card with a 78th-minute try from Gilem Queheille but Rokoduguni touched down with the last play, with Matt Banahan adding the conversion.
Newport Gwent Dragons missed the chance to finish second in the pool after failing to collect a bonus point in their 24-12 win at Italian champions Rugbby Mogliano.
Ross Wardle and Wayne Evans scored in the space of four minutes as Dragons were forced to overturn a 9-3 half-time deficit at Stadio Maurizio Quaggia.
Steffan Jones levelled the scores, with prop Nathan Buck in the sin bin, before the two tries and five points from replacement Kris Burton, but the Dragons could not grab a bonus even though Mogliano finished the game with 13 men.
Newcastle Falcons ended their Pool 3 campaign with a 28-0 win over Bucharest Wolves at Kingston Park but it was not enough to earn them a place in the quarter-finals.
Instead Brive secured top spot in the pool after beating Calvisano 31-9 in France.
The Falcons knew they needed a full house of points to keep up the pressure on Brive and Ryan Shortland gave them the perfect start with a try within two minutes of the kick-off. Rory Clegg kicked two penalties and converted a penalty try on the stroke of half-time for an 18-0 lead.
Wolves lost two players to the sin bin but it was not until the 74th minute that Tom Catterick crossed for the Flacons' third try that was too little too late.
Stade Francais ended London Irish hopes of progressing after winning Pool 5 by beating Lusitanos 48-15 in Portugal.