"We know what he is capable of and as a team we have a lot of confidence in him. He's still very young, but he is going to become a very good five-eighth," said Anscombe.
"He kicked his goals and our aerial game was pretty good. We got some good return from it, although we probably left a couple of tries out there.
"We don't seem to be having much luck at the moment but, provided we keep on working hard, that will change. The ball simply isn't sticking."
Anscombe, who guided Ulster into the PRO12 final and the Heineken Cup quarter-finals in his first season, now has his sights on picking up more points on the road in France in round two when the Belfast-based teams has to travel to high-flying Montpellier.
Travelling with a win behind them will make a big difference to the Ulster players'mindset and there was no feeling of upset at having conceded a penalty that led to the Tigers heading home to Leicester with something for their troubles
"They might be happy with a point but our objective was to get four points and we achieved that, each week and each game we play that is our objective and if we stick to doing that we'll come through," added Anscombe.
"We're happy with the four and there is only six games and you have to win your home games to be there at the end of the day and we have won our first one, we'll enjoy that and then we'll start to think about our trip to France next Saturday."