Ulster will now have to go to Leinster in the Pro12 semi-final while it is back to the drawing board for Ospreys, who have never previously failed to qualify for the top tier of European competition.
Ulster’s six tries came from outside-half Paddy Jackson, who provided a total of 21 points, hooker Neil Best, wing Andrew Trimble, flanker Chris Henry, replacement Stuart Olding, and lock Franco van der Merwe.
Opsreys scored four tries from outside-half Dan Biggar, centre Josh Matavesi, and replacement Tom Grabham (two) with Biggar providing three conversions.
The Ulstermen deserved their 21-14 half-time lead scoring three tries to two despite Ospreys having been awarded 10 penalties to Ulster’s nil.
Ospreys made their intentions known from the start by turning down kickable penalties in the pursuit for tries and the hope of a bonus point victory. And after 12 minutes the plan was going well when Biggar went in under the posts from a scrum with the conversion making it 7-0.
But Ospreys found themselves getting turned over too much in Ulster’s 22 and it was the visitors who struck with two tries in the space of 10 minutes.
Both scores came from scrums as Jackson was put over by full-back Jared Payne before skipper Best went in from 10 metres out. Jackson converted both scores as Ulster led 14-7 on 27 minutes.
But the home side got back on level terms when Matavesi reached out to score with Biggar converting.
It looked like the half would end 14-14 but a poor kick ahead by Rhys Webb was returned by Trimble and he got lucky with the bounce from his chip ahead to run in a final minute try. Jackson put over his third conversion.
It took until only the second minute of the second half for Ulster to open up a 10-point lead when Jackson converted their first penalty of the match.
The Ospreys did respond with an attack inspired by replacement Olly Cracknell but again they were guilty of turning over ball in Ulster’s 22.
On 56 minutes Henry scored an opportunist try by touching the ball down when it squeezed out of a ruck to give Ulster the bonus point try.
It was a controversial moment with the home crowd not liking the decision after Television Match Official Derek Bevan advised referee Marius Mitrea he could award the try. Jackson converted the try and then kicked a penalty as Ulster led 34-14 in the 60th minute.
Ospreys scored a third try when Grabham went over in the right corner but Ulster replacement Olding ran in a fifth try for the visitors just three minutes later with Jackson adding his fifth consecutive conversion.
Five minutes from time Grabham went over for his second and Ospreys’ bonus point try but it was too little too late. There was still time for Van der Merwe to crash over for try number six as Ulster finished with a flourish.
Tries: Biggar, J. Matavesi, Grabham 2. Cons: Biggar 3.
Tries: Jackson, R. Best, Trimble, Henry, Olding, Van der Merwe. Cons: Jackson 5. Pens: Jackson 2.
D. Evans, John, Watkin, J. Matavesi, Walker, Biggar, Webb, Smith, Baldwin, Arhip, Bernardo, Thornton, Lydiate, Underhill, King.
Grabham for John (60), Davies for Watkin (50), Leonard for Webb (69), Jarvis for Arhip (48), Bearman for Bernardo (58), Cracknell for Thornton (48), Otten for Underhill (64).
Payne, Trimble, L. Marshall, McCloskey, Gilroy, Jackson, Pienaar, Black, R. Best, Lutton, Browne, Van der Merwe, Henderson, Henry, Reidy.
Cave for L. Marshall (58), Olding for McCloskey (56), P. Marshall for Pienaar (72), McCall for Black (48), Herring for R. Best (67), Warwick for Lutton (64), Diack for Browne (56), Wilson for Henderson (56).
Marius Mitrea (Italy).