Clermont Auvergne came from behind to claim a bonus-point win and extend Bath's European Champions Cup misery.
Rhys Priestland's bright play saw Bath claim the first try but George Moala powered over to draw Clermont level.
Greg Laidlaw kicked two second-half penalties and two tries by teenager Samuel Ezeala came either side of an Alexandre Lapandry score.
Jonathan Joseph scored a late try as Clermont went top of Pool Three while Bath stay bottom on two points.
Bath fall short after bright start
The hosts needed a victory to breathe life into their campaign after defeat in the opening two matches, and they got off to a good start when Priestland impressively switched the direction of attack to hand his side the lead.
The Wales fly-half shaped up to take a close-range drop goal but instead motioned to run the ball wide before throwing a dummy pass to enable him to cross the line.
Bath grew confidence from taking the lead and continued to win front-foot ball that enabled them to play expansive running rugby.
Clermont - yet to win the competition despite playing in three finals - endured a difficult first half but drew level when the impressive Moala brushed off two tackles to score his side's opening try.
Clermont scrum-half Morgan Parra was replaced after failing a head injury assessment at half-time and his replacement, former Scotland captain Laidlaw, quickly got his side on the front foot with an early penalty.
Priestland levelled from the tee but Laidlaw edged the French side in front once again before three tries in seven minutes blew the Premiership team away.
Nineteen-year-old winger Ezeala was quiet for the opening hour of his European debut before bursting into life as he finished a well-worked move.
Moments later, the Barcelona-born back fed Lapandry to score Clermont's third.
The bonus point was sealed in the closing stages as Ezeala latched onto Camille Lopez's grubber kick to score the fourth try, before England centre Joseph crossed late on for the hosts.
'We let the details slip' - what they said
Bath director of rugby Stuart Hooper told BT Sport: "There was a big difference between the first and second half. That is where the frustration lies, because we let the details slip away from us and we gave a very good Clermont side opportunities."
On Bath's attacking intent: "That was specific to this game and sometimes we will play like that, but we let it get away and that is why the lads are hurting so much."
On the trip to Clermont next weekend: "For us, it is another opportunity to put on the Bath shirt [facing Clermont away next weekend]. As we always do, we will sit down and talk about selection and what that will look like over the next few months."
Line-ups
Bath: Homer; Rokoduguni, Joseph, Roberts, McConnochie; Priestland, Cook; Obano, Dunn, Stuart, McNally, Stooke, Williams, Bayliss, Louw (capt).
Replacements: Walker, Boyce, Judge, Garvey, Davies, Chudley, Burns, Brew.
Clermont: Abendanon; Ezeala, Toeava, Moala, Raka; Lopez, Parra; Uhila, Beheregaray, Slimani, Iturria, Timani, Fischer, Lapandry, Lee.
Replacements: Ulugia, Beria, Zirakashvili, Ruaud, Levave, Laidlaw, Betham, Naqalevu.
Referee: Andrew Brace (Ireland).