Rhys Gill, Ernst Joubert and Chris Wyles scored the tries and Owen Farrell, the 20-year-old who is in contention for a place in England's Six Nations squad, kicked 16 points.
The Ospreys kept themselves in touch with a brace of tries from Ashley Beck while Dan Biggar kicked the rest to ensure they left London with the consolation of a losing bonus point.
Saracens head into next weekend's rematch at the Liberty Stadium top of Pool Five with 10 points, three ahead of Biarritz and the Ospreys.
Saracens billed themselves as the wolfpack, running out to ominous chants of "big bad wolf" - and they lived up to the moniker right from the outset, tearing into their prey and taking a third-minute lead.
Charlie Hodgson fired a sublime long pass out to full-back Alex Goode, another England contender, who slipped the ball to wing Wyles on the outside.
The American was tackled into touch just before the line but Joubert stole the lineout and prop Gill drove over the line for the third-minute try, converted by Farrell.
Biggar and Farrell exchanged penalties before the Ospreys profited from a stroke of fortune to draw level with a breakaway try from Beck.
The Ospreys centre started the move with a grubber kick in behind the Saracens defence for Tommy Bowe, whose pass was tapped backwards by Hodgson. Beck scooped up the loose ball and accelerated clear of the broken Saracens defence to score under the posts, affording Biggar the simple conversion.
But Saracens hit straight back through number eight Joubert, who exploded off the base of a scrum on the 22 and through Tom Smith's attempted tackle to score.
Biggar landed a second penalty to keep the Ospreys in it but it was Saracens, driven forward by dominant back-row performances from Joubert and Jacques Burger, who were asking all the most potent attacking questions.
Hodgson was revelling in the front-foot ball, England prospects Goode and Farrell attacked from deep and David Strettle came close to scoring in the corner.
Ospreys captain Smith was sin-binned for persistent offending and Farrell landed two more penalties to stretch Saracens into a 23-13 half-time lead.
Biggar slotted the first points of the second period but Saracens were soon on the attack again.
Burger was involved twice as Saracens launched wave after wave of attacks on the Ospreys line before Schalk Brits created the opening with a delightful sleight of hand, slipping the pass behind his back for Wyles to finish.
Saracens were reduced to 14 men when Farrell was sin-binned for body-checking Biggar off the ball and the Ospreys capitalised, working a second try of the evening for Beck.
The Ospreys created a three-on-one simple overlap and full-back Richard Fussell committed his opposite number Goode before sending Beck over for the score, which was converted by Biggar.
Farrell made an inauspicious return to the action at fly-half, in place of Hodgson, missing a penalty shot and then having an attempted clearance charged down.
The Ospreys smelled blood but Saracens weathered the storm admirably and then turned the tables with a long clearance kick from Farrell and impressive chase.
Fussell was scragged and then panic set in. Biggar passed to Andrew Bishop but the centre lost his footing under the Ospreys posts and Jonathan Thomas had to hack the ball clear.
Saracens won a penalty from the subsequent five-metre scrum, which Farrell converted.
The Ospreys responded with a strike from Biggar which earned them the losing bonus point from a high-octane encounter.
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