Bowe scored his 29th touchdown in his 65th European Cup outing - but there was little else for the visitors to cheer as the French side won 41-17.
The Top 14 club made it a special day on their Champions Cup home debut at Stade Marcel Deflandre as they ran in five tries to take firm control of Pool One with a maximum 10 points out of 10 after two rounds.
The home side led 13-10 at the break, but then ran in four second half tries as they completely dominated the second 40 minutes.
Bowe’s consolation score came 10 minutes from time and allowed him to join Dafydd James on 29 tournament tries in equal fourth behind Chris Ashton (37), Vincent Clerc (36) and Brian O’Driscoll (32).
Bowe said afterwards: “It was very frustrating, especially the second half. We had to absorb so much pressure and they were the best side we’ve played this season.
“We felt at half-time the game was there for us but unfortunately the second half didn’t go the way we wanted.
“We knew it was going to be a hostile environment out there and that they are playing some great rugby.
“In the first half we took it to them and a few small errors by us let them in for a try.
“At times we showed what we wanted to do, but in the second half the pressure they put on us meant we had to defend and defend and defend.”
The 33-year-old Ireland wing continued: “We made a few errors and made it easy for them - and their quality shone through. Their size put us under so much pressure and they finished every chance they got.
“The relentless pressure they put us under meant we couldn’t get our hands on the ball. Our strength is playing expansive rugby, but we couldn’t get enough ball.
Ulster now have back-to-back games against Harlequins in December, but before then they have to meet Leinster in the Guinness PRO14 this weekend.
Meanwhile Irish assistant referee Joy Neville became the first female official to be involved in a Champions Cup clash when Exeter Chiefs made it back-to-back Champions Cup wins as they claimed a famous 27-24 victory away to French giants Montpellier.
In the other game in Pool 1, Nathan Hughes terrorised Harlequins as Wasps swept aside their Aviva Premiership rivals with a 41-10 bonus-point victory in a Champions Cup match in Coventry.
The England number eight spearheaded Wasps’ five-try assault on Quins as Dai Young’s men ended a miserable run of five straight defeats in style at the Ricoh Arena.
Tries: Sinzelle, Vito, Gourdon, Aguillon, Eaton. Cons: Lamb, Bales 2, Goillot 2. Pens: Bales 2.
Rattez, Lacroix, Doumayrou, Aguillon, Sinzelle, Lamb, Bales, Priso, Forbes, Atonio, Eaton, Nailiko, Botia, Gourdon, Vito.
Goillot for Bales (60), Corbel for Priso (52), Bourgarit for Forbes (64), Pelo for Atonio (56).
Tries: Lealiifano, Bowe. Cons: Lealiifano, Cooney. Pens: Lealiifano.
Piutau, Trimble, Cave, McCloskey, Bowe, Lealiifano, P. Marshall, Black, Herring, Herbst, Treadwell, O’Connor, Henderson, Henry, Reidy.
Stockdale for Trimble (52), L. Marshall for Cave (64), Cooney for Lealiifano (47), B