However it needed a penalty try in the last five minutes for them to emerge victorious and Anscombe knows they will have to improve ahead of a second straight match in Italy.
He said: "We have used a get-out-of-jail card. Let's be honest, we all know that, but we won.
"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.
"We were flat. The guys are normally fizzing and bubbling with energy and we were all but that.
"We have to look at this performance and ask how we can improve rather than forgetting about it.
"We have 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."
"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."
Anscombe has been able welcome back two key members to his side with Andrew Trimble and Paul Marshall both released by Declan Kidney for the game.
Trimble starts in a back three also featuring youngsters Peter Nelson and Michael Allen.
Marshall will link up with Niall O'Connor while Roger Wilson captains the team from No.8.
Treviso also welcome back a pair of internationals with Luca Morisi and Andrea Pratichetti both coming on to the bench.
They come into the game on the back of two straight RaboDirect PRO12 defeats to Glasgow and Connacht.
Coach Franco Smith said: "Ulster haven't lost yet this season and they are well organised.
"Their coach Mark Anscombe has done a great job to take on players who were already very good.
"It was clear that they never give up against the Zebre in a match they looked like they had lost.
"There are a very few tries who can win despite conceding four tries, especially only scoring two tries themselves."
At Stadio Monigo, 7.05pm local time (6.05pm UK & Ireland).
Referee: Peter Allan (SRU, 49th competition game)
Assistant Referees: Andrea Spadoni, Stefano Traversi (both FIR)
Citing Commissioner: Alberto Recaldini (FIR)
TMO: Stefano Marama (FIR)
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