Conor Chaplin’s 52nd-minute leveller was cancelled out by Benik Afobe two minutes later, after Lys Mousset had put the visitors in front early on.
The Blues fell behind after nine minutes as Tom Davies was beaten by Afobe and Mousset powered home the finish.
It looked like a long afternoon was in store for the hosts in front of a crowd of 7,182.
But Kenny Jackett’s side grew into the game against their Premier League opponents from Dorset.
The likes of Carl Baker, Kyle Bennett and Brett Pitman all had chances, while Afobe also whistled a shot wide.
It was Pitman who provided the ammunition for Chaplin to coolly level six minutes after the restart.
Afobe’s cheeky back-heel restored the Cherries’ advantage two minutes later, however.
Chaplin was an inch away from equalising with nine minutes left as he angled his shot off the post.
And the excellent Christian Burgess headed a late corner just wide on a useful afternoon’s work for Pompey.