A first-half try from Steve Borthwick, and late efforts from Matt Stevens, James Hudson and Danny Grewcock sealed a valuable bonus point.
Bath skipper Borthwick scored on the stroke of half-time to move his side 16-9 ahead.
And tries from Stevens, Hudson and Grewcock wrapped up the win, while Olly Barkley kicked 17 points.
Bath centre Olly Barkley:
"All we were concerned with in training this week was being relaxed and perhaps that showed for the first 40 minutes.
"We always believed we were in control and then the forwards started producing quicker ball in the second half.
"The more ball I get, the more I enjoy the game. It's a bit frustrating at times playing at 12.
"We said that if we didn't win today last week's victory in Dublin would have been a waste of time.
"We just coasted through the Heineken Cup last year, we never put in a decent performance and we wanted to do it differently this time around."
Bath: Best; Finau, Cheeseman, Barkley, Welsh; Malone, Wood; Stevens, Mears, Bell; Borthwick, Grewcock; Beattie, Scaysbrook, Delve.
Replacements: Williams for Best (79), Davis for Malone (49), Dixon for Mears (68), Loader for Bell (79), Hudson for Borthwick (75), Lipman for Beattie (60), Fea'unati for Delve (58).
Bourgoin: Peclier; Janin, Davis, Coux, Carmona; Boyet, M. Forest; Milloud, Cabello, Peyron; Pierre, Del-Fava; Diotallevi Baluc-Rittener, Jooste.
Replacements: A. Forest for Peclier (63), Giorgadze for Coux (63), Cardinali for Peyron (54), Mazel for Del-Fava (54), Monzeglio for Baluc-Rittener (54).
Not Used: Montagnat, Campeggia.
Sin Bin: Coux (33).
Ref: Nigel Owens (WRFU).