Lauri Dalla Valle fired Bournemouth ahead but Lee Barnard headed Saints level and captain Dean Hammond put them in front.
Rickie Lambert made the game safe with his 16th goal of the season three minutes from time before Saints had Oscar Gobern dismissed late on.
Adkins said: "There is still a long way to go. We have about a quarter of the season remaining and still have to play a lot of good teams. It is really tight at the top of the league with only a couple of points separating all of the top six.
"I am pleased with the three points. The game was very lively, which you would expect for a derby of this magnitude, but I did feel we let them into the game too much at times.
"This is the time of the season where it starts to get a little bit interesting, and we still have some really big games ahead of us."
Source: PA
Source: PA