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A try from replacement Francisco Leonelli and some top-drawer kicking from Glen Jackson saw off the Welsh regional side, who had been regarded as strong pre-match favourites.
Alan Gaffney's side will now face 2006 winners Munster at Coventry's Ricoh Arena having also avenged last month's 30-3 EDF Energy Cup mauling by Ospreys.
It was the visitors who took the lead after Richard Hill conceded a penalty right in front of the posts, which James Hook despatched in the third minute.
That advantage lasted just four minutes however when Jackson was presented with a penalty of his own from equally enticing range after Saracens' first attack of note was illegally stifled.
Jackson repeated the trick on the quarter-hour mark to nose the home side ahead but his third penalty attempt, from just inside the Ospreys' half, fell just short.
There was a nervous moment when an Ospreys clearance was charged down but that almost led to the visitors regaining the lead with the game's first try when Gavin Henson made an eye-catching interception.
That allowed Hook to make good ground but the pass that should have sent Sonny Parker sprinting for the line was clumsily dropped by the centre.
The near-miss sparked the game into life however and Saracens winger Kameli Ratuvou went on a powerful run that required some sterling defence to stop and that also applied to two forays at speed by Richard Haughton as the home side began to take control.
Misfortune also came their way however with two injury setbacks in the space of a couple of minutes as first Brent Russell and then Andy Farrell were forcedoff, with the latter clutching a shoulder and helped to the sidelines by two medics as play continued around them.
Lee Byrne's inability to grasp a pass on the left touchline ended a promising Ospreys move but the half ended with all eyes on Irish referee Alan Lewis as he awaited confirmation from colleague Brian Fitzgerald as to whether Paul Gustard's momentum had carried him over the line.
It had not but the home side were nevertheless good value for their slender half-time lead.
That was extended to 10 points within two minutes of the restart when Saracens' two replacements combined for the first try of the day.
The home side were initially defending deep but after Marty Holah got a touch to Neil de Kock's clearance to play Saracens onside, Adam Powell was able to feed Leonelli and the Argentina international knew he would score in the corner from the moment he evaded Ryan Jones' last-ditch tackle.
Jackson made a tricky conversion before Adam Jones knocked on within a few paces of the line from the Ospreys' next reply.
Gustard was forced off injured as another television decision went the defending Ospreys' way as bodies piled up on the line.
Byrne was sin-binned for slapping the ball away to stop Haughton going clear and Henson forced off injured before the boot of Jackson added another three points with his third penalty just before the hour mark.
Time was running out for Ospreys but replacement Paul James powered over for a try that was successfully converted by Hook and at 16-10 there was still a chance of a comeback despite just four minutes left on the clock.
Jackson had other ideas however and his smartly-taken drop-goal ensured it was to be his side's day.
Saracens (6) 19
Try: Leonelli
Con: Jackson
Pen: Jackson (3)
Drop: Jackson
Ospreys (3) 10
Try: James
Con: Hook
Pen: Hook
Yellow Card: Byrne (Ospreys)
Saracens: Russell; Haughton, Sorrell, Farrell, Ratuvou; Jackson, De Kock (capt); Lloyd, Cairns, Visagie, Vyvyan, Chesney, Gustard, Hill, Skirving.
Replacements: Ongaro, Johnston, Ryder, Barrell, Rauluni, Powell, Leonelli.
Ospreys: Byrne; Vaughton, Parker, Henson, Shane Williams, Hook, Marshall; Duncan Jones, Bennett, Adam Jones, Gough, Wyn Jones, Jonathan Thomas, Holah, Ryan Jones (capt).
Replacements: Hibbard, James, Ian Evans, Tiatia, Webb, Bishop, Brew.
Referee: Alan Lewis (Ireland).
