Sportal.co.nz 16.Dec.2013Getty Images
Anscombe's contract runs to the end of the 2013-14 season but he recently told the BBC that he wanted to extend his time with the province.
Anscombe said his job was the same as that of the players, to do what was expected of them.
"Mine [job] is to put out the best team week in and week out and achieve success and do well. If I do that and the players do that then people want you to stay around and do things," he said.
Anscombe said to stay he first had to be asked but he said he would like to stay.
There was a good environment and good people to work with, good players and some good talent coming through.
"We're not complete yet where we need to go but it's a pathway that we've started and any job you start you want to be able to finish it," he said.
The win in Treviso meant Ulster had achieved a first by winning their opening four games in the competition.
They have to wait until January for the next phase of the competition when they play Montpellier at home and away to Leicester on January 18.
In Saturday's win, one of the side's four try scorers was former Blues, Crusaders and Northland back Jared Payne.
But what pleased Anscombe the most was his side's ability to prevent the opposition scoring a try.
"Even when we had the game won, and they were on our goal-line, our guys fought and did not give them anything," Anscombe said.
Before contemplating their European games, the side face Pro12 matches against Zebre, Munster and Leinster.
"All the games we have coming up are against some of the higher quality sides in Europe and it gives us a chance to test ourselves and see how good we actually are.
"We would like to give a few guys a bit of a break without jeopardising the level of performance we expect and the standards we need to achieve.
"We are happy with where we are in Europe. The boys have put themselves in a good position.
"Their destiny in the Heineken Cup is in their own hands and that is what you want," Anscombe said.