The Ospreys bossed first-half possession but three Ruan Pienaar penalties gave Ulster a 9-6 lead at the break, before Matthew Morgan's third Ospreys penalty levelled the contest before a Stuart Olding penalty restored Ulster's lead.
However, a breakaway try by Ospreys prop Ryan Bevington proved the crucial score as the champions took victory.
Anscombe said: "That was pretty poor and I have just said to the guys that we seem to be saving our worst performances for our home track.
"It's a concern, the boys love playing here, we have the best supporters in the country and that is why I don't understand why we've produced that kind of performance.
"The Ospreys came here with a young team, they bullied us at the breakdown, they were more abrasive than us and they beat us up in the forwards.
"We had to live on scraps, while they took their opportunities and deservedly won the game.
"I'm not worried, we just need to front up now this week and analyse every aspect of the game.
"Our attack was too lateral at times and it was not threatening and we didn't ask any questions and that needs to change."
Ospreys coach Jonathan Humphreys was comparatively left beaming at the final whistle, he said: "I can't describe it, the boys worked hard, especially in the last two minutes - we were fantastic and I'm just so pleased.
"I thought the forwards were brilliant, absolutely brilliant - we had a lot of boys missing but those who stepped into the shirts ensured there was no difference in the performance.
"James King was phenomenal tonight but there were also others like Ryan Bevington and Joe Bearman that were also outstanding.
"I speak to Ryan Bevington all of the time and unless he ends up as the best loosehead in the world I will be disappointed.
"Physically the guy is immense and he possesses every attribute you need at this level. Against Ulster he was world class but I expect that of him and I know he can do that."
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