"We are smiling at the fact that we have four points and we are still top of the table, but we want to be better."
Analysing his team's performance, the New Zealander said: "We were flat. The guys are normally fizzing and bubbling with energy and we were all but that today.
"We just didn't do enough up front, the forward pack didn't provide the quality of possession that was needed. We lacked accuracy and Zebre grew in confidence and started offloading.
"Our kick receiving was very, very poor and they were hurting us badly. Every time that they kicked off, when we got points, we allowed them to get points back straight away. It all combined to make it a bad day at the office."
Looking for some positives, he added: "Character has been a trademark of ours this year and we showed that today. We hung in there and we played the 80 minutes to get the result."
He also found time to praise the Ulster contingent who played so well for the Ireland XV against Fiji on Saturday, with Craig Gilroy (3 tries), Darren Cave (1 try), Luke Marshall (1 try) and Paddy Jackson (5 conversions and a penalty) all getting on the scoresheet.
"The young players did well - they are all good footballers and it was good to see them getting their opportunity.
"They know that they have to keep their feet on the ground. It's early days in their careers and they still have work to do, but we know that in them we have got some good ones."
The Ulster squad will remain in Italy this week before taking on Benetton Treviso on Friday night. Anscombe is confident that his charges can respond in the right way and get the desired result.
"We have to look at this performance and ask how we can improve rather than forgetting about it," he explained.
"We have five days now to prepare for the Treviso game and we know that they will be harder again, so we have to lift it up for that game and fix what went wrong today."
The Benetton Treviso v Ulster match will be shown live via the red button on BBC 2 Northern Ireland, with the action getting underway at 6.05pm Irish time/7.05pm local time.