Dan McFarland’s side have won all three pool games and have 14 points out of a possible 15.
Ulster’s starting backline had an average age of 22 and it was the backs that did all the damage scoring all four tries.
Robert Baloucoune marked his return to the starting line up with a try on six minutes.
Marty Moore won a scrum penalty and Ulster kicked to touch, from clean lineout possession, Stewart Moore, James Hume and Billy Burns had probing runs, Nathan Doak tried to go through a gap but was stopped, his offload went to ground but Baloucoune picked up and had to strength to go over in the corner for an unconverted score.
Doak, on his first European start, got Ulster’s second try on 14 minutes.
Saints centre Rory Hutchison failed to gather the ball for an attacking kick deep in the Ulster half, Michael Lowry scooped the loose ball up and set off on a counterattack. He fed Baloucoune, the winger evaded a couple of tackles before popping it inside to Doak who had ran a great support line to go under the posts and the scrum half converted his own try.
Dan Biggar was on target with a 19th minute penalty for the home side.
Saints got back into the game three minutes before half time with a Biggar try on 37 minutes.
Following a turnover by Courtney Lawes, Biggar threw a looping pass to the wing, Hutchinson collected and galloped into the Ulster 22, he fed Tom Collins and the winger flung the ball inside for the Welsh outhalf to score and he converted his own try.
Ulster replied immediately. Burns’ chip caused havoc in the Saints defence, Lowry was the quickest to react to the loose ball, the full back picked up and went over for a try which survived a long check from the TMO.
Doak’s conversion gave Ulster a 19-10 lead at the interval.
Biggar landed his second penalty three minutes after the restart after they went off their feet in the shadow of the posts.
Lowry nearly got Ulster’s bonus point try on 55 minutes following an electrifying break, but he was stopped short, however it was deemed illegal and Saints scrum half Alex Mitchell was yellow carded.
Ulster wasted the opportunity as from a quick tap penalty as Eric O’Sullivan was turned over on the try line.
Lowry did get the bonus point try five minutes later. Baloucoune’s footwork beat two defenders and he took a third out of the game with his offload to Ethan McIlroy, the winger advanced before finding his full back, Lowry evaded the final cover tackle to cross for his second touchdown of the game, but Ian Madigan failed to convert.
Hutchison scored a late consolation try for the Saints which George Furbank converted but it was too little too late for the hosts as they exited the competition.
Ulster face Clermont at home on Saturday and will be looking to make it four wins out of four and guarantee one of the top seeding for the last 16.