Number eight Ernst Joubert capitalised on a charge down from a Dan Biggar clearance to cross for Saracens' try, with centre Owen Farrell adding the conversion and three penalties to seal victory for the Aviva Premiership champions, who ended the hosts' 17-match unbeaten European home record.
Biggar kicked two penalties for the hosts and converted a second-half try from lock Ian Gough, but they could not make their dominance of territory and possession count due to the visitors' phenomenal effort when not in possession, particularly during a second half where they played 20 minutes a man down after first Schalk Brits and then Kelly Brown were sin-binned.
The win kept Saracens top of Pool Five and their destiny will be very much in their own hands when hostilities resume in January, but the Ospreys, who now trail the English side by six points, face an uphill struggle if they are to reach the last eight.
As had been the case in last week's meeting at Wembley, Saracens made the better start and Farrell rewarded their early pressure by slotting two penalties from similar positions on the right, either side of a Biggar three-pointer for the home side.
The Ospreys enjoyed an early edge at the scrum, and referee Jerome Garces' decision to award them two penalties in that facet of play saw the front rows engage in a pair of minor scuffles as the game simmered despite the freezing conditions.
Biggar again levelled matters with his second penalty after 25 minutes when Saracens were needlessly caught offside in midfield, but that parity lasted less than a minute as the fly-half's attempted clearance was charged down by opposite number Charlie Hodgson, with Joubert the grateful beneficiary.
Biggar was off target with a straightforward penalty effort from 35 metres - an error that was compounded when Farrell rewarded a strong Sarries scrum by adding a simple third penalty.
The scrum tide had started to turn in the visitors' favour and things soon got worse for the hosts when Wales prop Paul James was yellow-carded for not binding, and they headed down the tunnel at half-time with their hopes of retaining their proud home record looking precarious thanks to a 16-6 deficit.
With that long unbeaten streak on the line, the Ospreys made a furious start to the second half.
Biggar missed a penalty before the home side thought they had found a way back into the game after sleight of hand from Richard Hibbard freed Ashley Beck. The centre stepped inside but was guilty of a double movement as he tried to force his way over the line.
With James restored to proceedings it became Saracens' turn to spend 10 minutes a man light as Garces harshly binned hooker Brits for an early scrum engagement.
Saracens survived that 10-minute spell but quickly lost Brown to the bin for a trip in Biggar, and this time they were made to pay when the fly-half worked lock Gough over in the corner and added the conversion to narrow the gap to three points with 14 minutes to play.
The Ospreys continued to batter away but, as had been the case throughout, they were stymied by errors forced by Saracens' suffocating tackling effort, exemplified by a man-of-the-match performance from flanker Jacques Burger.
Farrell missed a chance to extend Saracens' lead as he pulled a penalty attempt after a rare second-half venture into Ospreys territory, but it did not prove costly as the visitors saw out the final five minutes to claim the spoils.
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