"It was nowhere near good enough," said the 23-year-old, after steering the Swansea-based team to their third win in nine days.
"We were nowhere near what we've been the last three or four weeks and we can only apologise for the performance we put on. The coaching staff have put a lot of work in with us and we let them down over the 80 minutes.
"Credit to Zebre, I thought they played very, very well, were very competitive at the rucks and we just didn't deal with it. If you don't deal with the ruck area in this game, in this day and age, it's an uphill battle.
The home side were in danger of becoming the first team to lose to Zebre in the RaboDirect PRO12 as they went in at half time trailing 8-6. But Biggar struck a penalty and then converted his own try to give the Ospreys what appeared to be a decisive lead.
However, Tito Tebaldi touched down for Zebre to put the result in doubt and set up a nervy final few minutes for the Ospreys, who hung on to claim their ninth win of the season.
But Biggar, who also struck three penalties, knows a large improvement is needed if the Ospreys are to get anything out of their Heineken Cup game with Leicester Tigers at the Liberty Stadium on January 13.
"We didn't help ourselves either with silly knock-ons, just going away from our pattern a little bit, but they really did come hard at the ruck area and caused us a lot of problems," he added. "Leicester will do exactly the same next week so we're in for a hard week's training and we can't complain about it.
"At the end of the day we've won a game. We're happy with the result all in all, after that performance, but we know that we're going to have to play an awful lot better, not just next week but the rest of the season if we're going to finish in the top four places.
"We've down our bit, just about, the top four is where we need to be and hopefully now we can stay here. We've had a good Christmas all in all. We're obviously disappointed with the way we played tonight but three wins in nine days is good return."