Brett Pitman gave the Cherries an early lead but Jimmy Phillips equalised for Albion and it looked like the two sides would go into half-time level. However, Tommy Elphick restored the hosts' advantage in the 45th minute.
In the end the Championship side eased to a 4-1 victory that probably only just flattered them, such was their attacking prowess, to set up a lucrative home tie against Liverpool in round four later this month.
"Bournemouth are a terrific side and it's a good lesson. If we want to be a better side we've got to take one or two things out of that game," said Rowett.
"We've had some really good results in the cups against good sides, and I was hopeful that we could have made the game a little bit more difficult for Bournemouth.
"But if a team are on it, and they're better than you, then the reality is, they're going to beat you.
"(Bournemouth manager) Eddie Howe has created such a good team here, and the pace and athleticism they showed was a real lesson to us. If that's what we take out of the game alone, then it will be worthwhile for us."
Rowett's side held the Cherries to a one-goal lead until the 85th minute but in the final five minutes, Ryan Fraser and Pitman again made the scoreline more resounding.
- Sports & Recreation
- Brett Pitman