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Ulster, who hadn't lost a game in any competition this season before this defeat, are still the Pool leaders, but only have the benefit of a three-point cushion between themselves and Castres Olympique in second place.
Saints are five points behind in third place but all three sides will have very real hopes of their own of making it through to the knock-out stages and a lucrative quarter-final place.
South African No 8 GJ Van Velze's try proved the difference between the sides but the chase for a quarter-final race is still a three-way race between Ulster, the Saints and Castres.
The French side were pushed all the way by a battling Glasgow Warriors side, who cannot qualify for the quarter-finals, and only just managed a narrow 10-8 home victory at Stade Pierre Antoine.
The Scottish side, who have failed to win a game in the Pool so far, saw a try by flanker Yannick Caballero on the 73rd minute rob them of a famous win on French soil.
Fly-half Ruaridh Jackson saw his late penalty on the 77th minute hit the posts as Glasgow snatched defeat from the jaws of victory against a physically intimidating Castres line-up
