Stand-in skipper Rob Herring celebrated his new leadership role with the opening try as Ulster eased to comfortable but unimpressive win.
Ulster will be delighted to have opened up with a bonus victory after further tries from Wiehahn Herbst, Luke Marshall and man of the match Stuart McCloskey, who also added two penalties and a conversion.
Ulster started confidently, with young fly-half Peter Nelson showing few nerves despite wearing the No 10 jersey for the first time in a senior game.
The hosts went ahead on six minutes, with Nick Williams flipping a superb pass out of the tackle to put returning flanker Willie Faloon away on a gallop deep into Ospreys' red zone. A quick recycle allowed skipper Herring to stretch over.
With no recognised kicker in the side, centre McCloskey made a hash of the conversion but slotted over a penalty three minutes later.
Following his eight-match ban for foul play at the tail end of last season, Williams was making up for lost time, making big hits, linking and generally making a nuisance of himself.
Ospreys finally got to see some action in the Ulster half, but a stoic defence held firm, restricting the visitors to a penalty from Sam Davies.
After a spell of lethargy, Ulster bounced back with a brilliant midfield break by discarded Ireland wing Craig Gilroy, and even better thundering run from McCloskey as he crashed over with three defenders hanging on to him. He celebrated by converting from the touchline.
Davies landed his second penalty which cut the lead to 15-6 at the break.
After the interval, the impressive McCloskey smashed through the Ospreys defence to earn a penalty which he kicked.
After Williams was again involved in a series of drives, tight-head Herbst drove over in the corner.
Although the Ospreys dominated possession in the fourth quarter, Ulster swooped in at the death for that bonus try through Marshall.
ULSTER - L Ludik; A Trimble (D Shanahan 67), L Marshall, S McCloskey (S Arnold 60), C Gilroy; P Nelson, P Marshall (J Andrews68); A Warwick, R Herring (capt), W Herbst (R Lutton 60), D Tuohy (Browne 54), F van der Merwe; R Wilson (C Ross 3-14, 33-40 blood), W Faloon (S Reidy 66), N Williams (C Ross 56).
OSPREYS - D Evans (S Underhill 63); K Phillips, J Spratt (A Beck, 48; S Otten 66), B John, T Grabham; S Davies, T Habberfield (B Leonard 63); M Thomas (N Smith 52), S Parry, D Arhip (C Griffiths 64), L Ashley (capt), D K Steenkamp (R Thornton 54), J Bearman, L Evans, G Delve (D Baker 48).
REF - I Davies (WRU).