Steve Tandy’s men edged into a 12-6 lead at the interval thanks to tries from Eli Walker and Justin Tipuric.
But they put the result beyond doubt and wrapped up a maximum points haul with further tries from Jonathan Thomas and Kahn Fotuali’i.
Walker opened the scoring for the Ospreys following an uneventful opening quarter. Ashley Beck broke the visitor’s defence and Kahn Fotuali’I went close before the ball was recycled and spread right for Walker to cross the whitewash.
Biggar converted but former Cardiff Blues fly-half Dan Parks chipped away at the Ospreys’ lead with a pair of penalties.
But the Ospreys finished the half on a high and extended their lead. Richard Fussell was once again the provider, he made a silky break before sending Tipuric over in the corner.
Biggar was off target with the conversion but the try had handed the Liberty Stadium outfit a slender 12-6 lead at the interval.
And the Ospreys quickly got the scoreboard ticking once again following the break as Connacht did themselves with no favour with Thomas picking off Parks’ pass to score.
Parks made some amends with his third penalty to bring the scores to 19-9, but from there it was one-way traffic.
The Ospreys almost grabbed the bonus-point in emphatic fashion as Wales Under-20s star Eli Walker ran riot. But his pass to Tipuric was miss-timed and the move eventually broke down.
Steve Tandy’s men continued to lay siege to Connacht’s try line and twice they kicked to the corner from penalties only to be frustrated.
But they secured the maximum points haul in the final ten minutes. Following a succession of scrum resets, Connacht were penalised and Fotuali’I bulldozed over from a quick-tap.
