Albion extended their unbeaten run to four games with a comfortable victory to progress to the fourth round, thanks to goals from Anthony Knockaert, Yves Bissouma and Florin Andone.
Chris Hughton said he was 'really delighted to get through', but admitted they were fortunate to be ahead at half time.
He said: "I'm really pleased. It was a difficult game as we expected. We didn't start the game too well.
"Bournemouth played very well. They were very expansive in a very high tempo game. We had a few problems and were probably 2-0 up against the run of play.
"Sometimes it happens that way and the pleasing thing for us is that we are getting into the positions to test the goalkeeper. We managed the game well in the second half, We knew we would need another goal because this is a Bournemouth side which has goals in them."
Sitting in 13th place, 10 points clear of the relegation zone in the Premier League, Hughton said he would like to have a strong cup run this year.
He added: "It was a good performance and that's exactly what they needed.
"It always puts a marker in my mind when players come in, in this type of game against very strong opposition, when they haven't played as much football as they might have liked, but still put in good performances.
"We've got a tough period coming up, in particular the next two games which don't come any tougher than that [Liverpool and United]. I would rather be in the next round than not."
Hughton also reserved praise for stand in goalkeeper Jason Steele.
"He made excellent saves, particularly in the second half. We knew he would be tested. Bournemouth when they have to go for it can be at their most dangerous.
"Thankfully he (Steele) was there to make the saves that he needed and we needed to make sure we were defensively strong as possible. We had really good opportunities to break."
Hughton admitted that the ideal draw would be to get a 'home draw against a team not in the Premier League'.
"That's not being disrespectful to anybody," he stressed.
"These competitions are very important to me and the club. The modern trend is to change things around. As we don't have too many injuries, we are in a position where anyone who comes in has played quite regularly.
"It's a great competition and hopefully we get a good draw."