A bonus-point victory at Stade Ernest Wallon took Bath to within a point of previously unbeaten Toulouse, setting up a fascinating finale in the sixth and final round of matches next weekend.
Toulouse have 16 points, Bath are on 15 with Glasgow, who overcame Montpellier 21-10 on the back of three tries from Canadian DTH van der Merwe, on 14.
The five pool winners and three best-placed runners-up qualify for the last eight. Bath host Glasgow next Sunday with Toulouse away at winless Montpellier.
Bath, second in England's Premiership behind Northampton, lost their first two matches in the new European Champions Cup but three successive wins have given Mike Ford's side momentum.
Matt Banahan, Chris Cook and Jonathan Joseph all crossed in the first-half as Bath raced into a 25-7 lead. Yoann Huget added to Yoann Maestri's first try for the hosts but Francis Louw scored Bath's fourth try 10 minutes from time, with England flyhalf George Ford adding 15 points with the boot.
Wales international George North got one of Northampton's two tries in a 20-9 win at Ospreys to set up a Pool 5 decider next Saturday against Racing Metro at Franklin's Gardens.
Racing, who walloped Treviso 53-7 at home, and Northampton have 19 points with next week's winners going through, although the losers would still be well placed to qualify as runners-up.
(Writing by Justin Palmer; editing by Martyn Herman)
