Eunan O'Kane's 53rd-minute strike was enough to secure the Cherries a place in the second round for only the second time in eight years and local bragging rights over their League Two neighbours.
Howe said: "I'm delighted to be in the hat for the next round because these games against lower league clubs are never easy and Portsmouth are a good side.
"We have dreadful record in this competition so hopefully we can go on a run now. We would relish playing a Premier League side in the next round."
O'Kane gobbled up the rebound after a Ryan Fraser shot had been pushed out by Pompey goalkeeper Phil Smith.
Wes Thomas should have opened the scoring for the home side midway through the first half when he blazed a header over from Simon Francis' cross.
Championship Bournemouth would have won by more but for a string of fine saves from Smith.