Bath left it late, winger Will Muir scoring the decisive try in the closing minutes to seal a 29-25 bonus-point home win.
The victory nailed down Bath's place in the last 16, while a Racing team featuring South Africa's World cup-winning skipper Siya Kolisi need to beat Cardiff next weekend to give themselves any chance of qualification.
Nolann le Garrec and Thomas du Toit scored opening tries for the respective sides.
Racing built on their 10-8 half-time lead with further tries from Fijian No 8 Kitione Kamikamica and English winger Henry Arundell.
But the hosts refused to roll over, Alfie Barbeary and Joe Cokansiaga crossing for tries before Muir's coup de grace eight minutes from time for a fine comeback victory.
