Ospreys' European hopes were dealt a blow as Racing 92 came from behind to win their Champions Cup clash.
Dubbed the "battle of the budgets", Ospreys defended heroically against the French giants with Keiran Williams' converted try giving them a half-time lead after Juan Imhoff's score.
But Racing found another gear with tries from Virimi Vakatawa and Teddy Thomas.
Racing remain top of pool A while Ospreys are bottom.
The top eight in each two pools progress to the last 16, with Ospreys due to travel to Sale on 23 January, while Racing host Northampton.
Ospreys went into the game as underdogs, with head coach Toby Booth pitching his side against the Galacticos of Paris, and they were dealt an early blow when Alex Cuthbert was shown a yellow card for a poor high tackle on hooker Teddy Baubigny - not the return to the starting XV the Wales wing would have wanted.
Maxime Machenaud, who assumed kicking duties ahead of Finn Russell, put Racing in front with the resulting penalty.
Ospreys then endured 10 minutes under the cosh, but defended aggressively to keep Racing at bay, the highlight coming when Williams won a turnover.
Williams, enjoying his first start since October, repeated his heroics at the other end of the pitch shortly after as Ospreys returned to full complement.
The centre powered over after some good work by the forwards in their first visit in the Racing 22.
Anscombe kept his cool to slot over the conversion in the face of Thomas' charge down.
Pacey backs Luke Morgan and Matt Protheroe were then left to rue two further chances as Ospreys grew into the game, but their handling let them down.
The home side got their just rewards through a long-range Anscombe penalty.
It took half an hour for Racing to get the try they had been threatening, the ball shipped wide to Thomas, who showed fine football skills in his chip and chase, before Pumas winger Imhoff dotted down.
Ospreys held on to a two point lead at the break thanks to Machenaud's off-target conversion attempt.
In a mirror of the first-half, Racing started with tempo with Ospreys once again reduced to 14-men. Will Griffiths was, perhaps harshly, shown to the sin bin for a deliberate knock-on.
A relentless Racing, playing Barbarians-style rugby, threatened the try-line as Russell began to work his magic, but Ospreys again defended heroically.
Anscombe attempted to relieve the pressure with an ambitious half-way penalty attempt, but it dropped short for the visitors to launch another counter attack.
Man of the match Gael Fickou broke the line before picking out Vakatawa, who would not be stopped in support. Replacement scrum-half Antoine Gibert added the extras.
Racing looked to have scored again soon after from a driving lineout, but Camille Chat was deemed held up over the line after consultation with the TMO.
The one-way traffic continued even when Ospreys were back to 15.
Russell stepped up as their third place kicker of the afternoon and extended their lead to two scores with a successful long-range penalty.
Thomas made certain of the result, running a try in off the wing after a looping pass from Australia's Kurtley Beale, but time ran out as they searched for the bonus-point.
Ospreys captain Rhys Webb told BBC Sport Wales: "We were proud of the first 40 minutes, we put a good shift in but we have to play a full 80 minutes to compete against these teams in Europe.
"We brought the intent but we've got to have the accuracy as well and we didn't have it at times, we left them to control the game at the end.
"We've got to get back on the horse ready for Sale away, it's a huge challenge and we had opportunities when we played them at our place, so it's something the boys are excited about."
Ospreys: M Protheroe; A Cuthbert, O Watkin, K Williams, L Morgan; G Anscombe, R Webb (capt); G Thomas, D Lake, T Francis, R Davies, A Beard, W Griffiths, J Morgan, E Roots.
Replacements: E Taione, N Smith, M Fia, W Hickey, S Cross, R Morgan-Williams, J Thomas, M Collins.
Racing 92: K Beale; T Thomas, V. Vakatawa, G Fickou (capt), J Imhoff; F Russell, M Machenaud; E Arous, T Baubigny, A Oz, L Jones, B Palu, I Diallo, B Chouzenoux, A Hemery.
Replacements: J. Maïau, G. Gogichashvili, C. Gomes Sa, V. Moreaux, C. Chat, A. Gibert, Olivier Klemenczak, D. Taofifenua.
Match officials
Referee: Christophe Ridley (England)
Assistant referees: Dean Richards & Wayne Falla (England)
TMO: David Rose (England)