A Fabian Schar penalty, won by Miguel Almiron, was enough for Newcastle to win 1-0 on Tuesday night.
Afterwards, Howe said: "Job done, through to the next round. These are difficult games but I was frustrated with our attacking play. We were not dynamic enough. Credit to them for their defending.
"The one on Miggy looked like he was caught. It is instantly forgettable. It won't live long in the memory. It is wins like this that create a cup run. You have to battle your way through."
Wimbledon boss Johnnie Jackson, meanwhile, was furious with the penalty.
Jackson said, "We have lost by a dodge penalty. I am fuming with it. First there was a dive (when Schar went down), free-kick, yellow card. Then there was another coming together, ref didn't give it then the linesman got involved and gave a penalty. I am angry about that one."
He continued: "It was a brilliant effort from our players. The way we did our game plan Newcastle and they know they have been in a game.
"The gulf in divisions and individual qualities is obvious but we bridged it and frustrated them. I am disappointed we did not force it to pens. I think we are a really good team, They are in there gutted we did not pull it off."