Ireland and Leinster fly-half Jonathan Sexton kicked six penalties - three during a 15-minute spell in the second period - while his two clinching goals arrived during the closing seven minutes on Sunday.
Before then, Bath had been close to claiming a notable scalp, England wing Matt Banahan having barged over from close range and Olly Barkley booting the conversion to put them in front after 65 minutes.
Barkley also kicked two first-half penalties but the effort was not enough as Sexton held his nerve towards the end.
Bath now head to Dublin for next Saturday's return fixture and, while they are not mathematically out of quarter-final contention, it will require a Herculean effort to turn things around, with a losing bonus point providing scant consolation.
In the same pool, Glasgow Warriors maintained their pursuit of Leinster with a 20-15 victory over Montpellier at Firhill.
In what as an attritional affair, Duncan Weir laid the foundations for the win, kicking four penalties before Ruaridh Jackson came on for the final 23 minutes.
He slotted a fifth pentalty with 12 minutes left before Federico Aramburu sealed the win by going in for a try.
Montpellier's tries came from Eric Escande and Masi Matadigo, the latter's effort coming in the last minute to earn his side a losing bonus point.
Leicester surrendered their unbeaten record in this season's tournament, and their position at the top of Pool Four, to Clermont Auvergne as they lost 30-12.
Leicester, who struck through a penalty try and Ben Youngs' effort, were second best for most of the game and now need to win the return clash at Welford Road next weekend.
The Tigers, who had won their previous two pool games against Italian outfit Aironi and Ulster, paid a heavy price for their own indiscipline.
Hooker George Chuter and England centre Manu Tuilagi were both sin-binned in the 50th minute for dangerous play, leaving the Tigers with 13-men for 10 minutes.
Clermont, with Julien Malzieu and Wesley Fofana (2) scoring the tries, made the most of their two-man advantage to record their 35th consecutive home victory in front of a partisan 18,000 crowd at the intimidating Stade Marcel Michelin.
It was not a good weekend for the English sides.
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