Piennar would add another two penalties in the second half to help grind out the 13-6 victory in what was a windy and wet Friday night at Ravenhill.
The game brought an end to a run of four games against Italian sides, with their last two matches seeing them beat Benetton Treviso in the Heineken Cup.
But while their defence has now not conceded a try in almost a month since their 19-11 RaboDirect PRO12 win over Zebre, Anscombe challenged the team to turn in a better 80-minute performance.
"We dodged a bullet there," Anscombe told the Belfast Telegraph. "We didn't play well but we'll bounce on and we'll now have to lift the bar.
"That was a tough game and the fact is that getting up and performing the week after the Heineken is traditionally tough.
"Even when you make changes we have enough experience here to demand a bit more of ourselves.
"It's a process we've got to look at it but it was a poor performance, but we got the win.
"I thought the defence there at one stage before half-time was pleasing and I think it gives us 267 minutes now without a try scored against us so that's pleasing.
"And that's what you want and the boys fought on the goal-line just before half-time but that attitude we showed there for five minutes wasn't the attitude that we showed for the other 75 or so."
The match also marked Paddy Wallace's return to the first team since suffering a serious knee injury last February.
The 34-year-old is the club's record appearance maker with 189 games in an Ulster shirt and Anscombe believes the playmaker can have an impact going forward.
"It's been a long year for him and he'll be better for his 60-odd minutes," he added.
"He's still got a lot to offer us and it's good that he's got the confidence now with a few games under his belt. With a few more he'll be better for it."
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