Goals from Lewis Grabban and Marc Pugh, either side of Lowe's red card for a reckless lunge, plus Harry Arter's late stunner handed the Cherries their third straight win and condemned the Dons to a fourth defeat in six games.
Bournemouth were the better side from the start and Grabban put them ahead after nine minutes when he intercepted Jon Otsemobor's weak back-pass and rounded goalkeeper David Martin to slot the ball home.
Lowe was shown a straight red card in the 45th minute for a challenge on Tommy Elphick after the pair had squared up minutes earlier.
Pugh doubled the Cherries' lead in first-half stoppage time, firing a rebound high into the net after Martin had saved Grabban's low strike.
The Dons showed plenty of spirit after the break and Shaun Williams curled a long-range shot against the post.
But Bournemouth were always in charge and Grabban was denied in one-on-one by Martin before Arter lashed a stunning 25-yard drive into the top corner to seal the visitors' win six minutes from the end.