Although dominating possession, Ulster failed to make their chances count in the final third, an aspect of their play Anscombe will look to improve in the weeks ahead.
"That's what we are here for, to win our first game and we achieved that. I wouldn't think any team will be singing their praises, thinking they've got everything right in the first round of the competition.
"Our objective tonight was to win at home and as you'd expect we've got things to work on, but a lot of positives as well."
Ulster led 15-3 despite replacement out-half Niall O'Connor missing four place-kicks. The gap was back down to just five points after Glasgow winger Tommy Seymour finished off a fine individual try.
"We got out to the lead and we were still pushing and trying to play rugby which was pleasing. I don't think our defence was good," admitted the New Zealander.
"We got guys on the deck a bit too much at times and didn't bounce up and spread our D (defence). I don't think they were hurting us. There was the odd missed tackle but we kept Glasgow at bay pretty well."
Although indicating that Andrew Trimble and Stephen Ferris could return to face the Ospreys next week, Anscombe stated that the players he has placed his trust in thus far have hardly put a foot wrong.
"It is about what the boys can do at training and conditioning them. There is two parts to the conditioning. The work they can do as far as getting their fitness up to speed and then it's getting them into game conditioning.
"Taking some knocks, getting into the contact aspects of training. It's really case by case of how advanced each player is.
"No matter who you are and how good you are, very few, if any in the world, are at their top game if they haven't been playing for two or three months.
"We've got to make sure and look at the ones, when they become available, what work and what content of workload they've done during the week to think if they can challenge to be put into the 23," he added.