The visitors started brightly and took the lead on three minutes when James Hook slotted home a penalty. But the hosts hit back and levelled the tie on 10 minutes thanks to Glen Jackson's opening penalty.
Jackson followed up with a second successful kick five minutes later to put Saracens three points clear. Lyn Jones's men looked threatening in possession but failed to take their chances by wasting a host of golden opportunities in the opening quarter.
Gavin Henson's neat interception on 22 minutes seemed to spark the visitors into life and Hook's pass to Sonny Parker should have resulted in the first try of the match. But the Wales centre dropped the ball with the try line at his mercy.
The Ospreys then spurned another chance when Jonathan Vaughton spilt the ball. And with the home side growing in confidence despite injuries to Brent Russell and Andy Farrell, the Ospreys were pressed on the back foot.
Saracens' domination nearly clinched the rewards in stoppage time of the first period. Flanker Paul Gustard scrambled for the line but was held up by the combined efforts of Lee Byrne and Shane Williams.
But they weren't to be denied as two minutes into the second half, Francisco Leonelli took advantage of Jackson's cross kick and beat the challenge of Ryan Jones before touching down in the right corner. Jackson converted to hand Saracens a healthy 16-3 lead.
The Ospreys rallied through Paul James's late try but Jackson's last-gasp drop goal clinched a deserved victory for Saracens, who reach the semi-finals for the first time in their history.
Click Here for a Statistical Match Report from the Heineken Cup Quarter-final at Vicarage Road