"Easy? You think it's easy?," said Anscombe after the three-try success at Ravenhill. "I don't think it's easy. I don't sleep some nights and that's not easy.
"It's challenging and that's what coaching is when you are doing your job. We are not fooled by what we've done, that it is how it's going to be."
Despite the win, Ulster's kicking looked particularly shaky early on with some aimless box kicking and Paddy Jackson missing two kicks just to the left of the posts.
While conceding that their kicking was 'pretty average' in the first 30 minutes, Anscombe was pleased with how their confidence grew with Springbok Ruan Pienaar showing his class as the game wore on.
"Ruan, who is arguably the best kicker/scrum half in the world, got his kicking game going and I thought the chase by (Andrew) Trimble and (Craig) Gilroy was pretty good. That put pressure on them and we started cramping them down in their own half."
Ulster well and truly had the upper hand in the set piece. Their scrum was hugely effective and they took a 13-9 lead just before half-time when referee George Clancy awarded the hosts a penalty try.
"Our scrum was outstanding and the key to our victory, without doubt. Robbie Diack did a great job for the lineouts and that gave us a good platform," admitted Anscombe.
"We were a bit tentative in the first 20 to 25 minutes and looked as though we didn't quite believe in ourselves against them. As time progressed we built self belief and things started to happen, confidence grew and I think we controlled the game in the end."
Despite missing the likes of captain Johann Muller, Dan Tuohy and Lewis Stevenson from the pack, the pleasing aspect for Anscombe was that the replacements stepped up to the challenge of meeting the reigning Heineken Cup champions head on.
While still in control of their season and putting 16 points between themselves and Leinster in the league table, the Ulster boss hopes to reach the heights of fellow New Zealander Joe Schmidt and Leinster in the second part of the season.
"If you look at what Leinster have achieved and done over the last couple of years, I tell you we would be pretty happy if we could achieve half of that, that's for sure.
"We've had a good result. We didn't have one last week (against Northampton) and there will be more of them across the year.
"It is how we bounce back, how we respond and how we challenge ourselves to grow as a unit. So hopefully, we'll be in most competitions come the end of the season, but there is a lot of water to go under the bridge before then."