Robbie Diack grabbed two tries and Rob Herring and Ricky Andrew one each in the bonus-point win, with Ruan Pienaar converting three while Daniel Halangahu kicked a penalty for Zebre's sole score.
However Ulster coach Anscombe was less than impressed with the performance and believes national commitments will leave them under strength for their top-of-the-table clash with Glasgow.
Ulster travel to Glasgow, who beat Newport Gwent Dragons 60-3, on Friday two days before Ireland's Six Nations clash with Scotland with Anscombe worried he may be left short for the fixture.
"With Ireland playing on Sunday and us on Friday night, we may not get anyone back from the National Training camp and that could mean five players not getting any game time at all next week," Anscombe told the Belfast News Letter.
"It is a matter of getting through this international window and surviving, not losing too much. Glasgow certainly are on fire at the moment, they are the form team in the league.
"I think the only thing that was better this week was the result. We got a bonus-point try, but I think our play was poor. It was horrible at times.
"We turned the ball over 20 times in the game and it is a trend we do regularly and I just think we are not getting any better at it and that is a concern.
"It is not a lack of commitment. That was good out there. The guys trained well but it is just option taking and decision making under pressure which let us down too often."
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