The Welsh region touched down three times in the opening 29 minutes, including an intercept effort from Dan Biggar, with young out-half Craig Ronaldson keeping Connacht in touch at 22-12 down at half-time.
It was a two-point game after Aly Muldowney set up Danie Poolman for Connacht's opening seven-pointer but even though Kieran Marmion added a second, further tries from Tito Tebaldi, Jeff Hassler and captain Alun Wyn Jones made it a 17-point winning margin in the end.
Ronaldson was making his first start at this level after a stellar season for Ulster Bank League club Lansdowne, and he settled any early nerves with a fourth minute penalty.
The 23-year-old missed his second shot at the posts and it was his pass which opposite number Biggar read for his intercept try, the conversion moving the visitors 7-3 clear against the run of play.
The Ospreys, with their strong pack growing in influence, touched down again on the quarter hour mark.
Ian Evans and the returning Justin Tipuric were both prominent in the build-up, exposing some poor tackling from Connacht, and full-back Richard Fussell provided the finishing touches with Biggar converting.
Steve Tandy's men might have had a quick third try as second row Evans bounded through once more, however he threw a loose pass to Hassler which the winger knocked on.
Connacht were unable to pressurise the Ospreys' try-line and were largely reliant on the left boot of Ronaldson, who tagged on two more place-kicks before he hobbled off to be replaced by Dan Parks.
By half-time the Ospreys had run in a third try through Eli Walker, who made it over in the left corner following a pacy break, and Biggar kicked one of two penalty attempts - the second hitting the post late on.
Ten points adrift, Connacht stepped it up a gear on the restart and Matt Healy went close to scoring before fellow winger Poolman broke up a listless spell with his try, gathering replacement Muldowney's pass out of the tackle to score by the posts.
The Ospreys always seemed to have an extra gear when required and their Italian scrum half Tebaldi sold a couple of dummies to grab a 'soft' try from Connacht's perspective. Biggar converted for a 29-19 scoreline.
Connacht responded with Marmion's effort, which owed much to Poolman's bright build-up play, but the Ospreys stamped their authority on the game during the concluding 18 minutes.
The composed visitors sealed the result with a spell of strong running and crisp passing that sent Hassler over for the clinching 71st minute score.
Biggar then completed his 18-point haul which makes him the first Welshmen to score over 1,000 league points. He added the extras to his skipper's easy finish, the final try being created by a quick free-kick and carry from replacement Richard Hibbard.