The Ulstermen had to remain disciplined and clinical throughout in order to secure a 20-19 win and have still only lost once in the last 12 months at home.
And while Anscombe, who replaced Brian McLaughlin this summer as head coach, admitted that while his players deserved a lot of credit for their start to the season, there was still much room for improvement.
"When you beat a team like Munster, you have to enjoy it. It is important to enjoy your wins," he said.
"Yes, we know we have a lot of work to do, but you cannot beat yourself up about that immediately after you have beaten a top class team like Munster.
"Yes, I was very worried after the first half, I was worried in the last second of the game. To be honest I do not think there was one minute of those 80 that I was not worried.
"Two weeks in a row we know we have character in this group. That is very clear to see. We know we have guys in there who have got something about them.
"We will enjoy that victory, but we know there is a bit of work to still be done. We still have players to come back in and we still have to build.
"There are things we are still not getting right. We are where we are, but this is only the third game of 22 league matches. There is a hell of a long way to go yet and we are not getting carried away by it all."
More than 10,000 fans packed into Ravenhill on Friday night and except for a defeat to European champions Leinster back in April they remain unbeaten there in the best part of a year.
And Anscombe paid tribute to the supporters who cheered his troops to a dramatic win.
"The crowd was outstanding and it was great to experience that and I think they have to take credit for taking us home in that last five minutes," he added.
"They got the boys home, they were outstanding."
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