Mark McGhee's side have not won away since November but there is nothing wrong with their home form. They romped to a fifth win in seven games at Withdean with three goals in the opening 24 minutes.
McGhee made four changes following the 2-1 defeat at Grimsby and his selection soon paid off.
The recalled Gary Hart finished off a fine run by smashing in his first goal since September in the sixth minute.
Trevor Benjamin, beginning a second month on loan from Leicester, doubled Brightons lead seven minutes later. The big striker latched onto a terrible back pass by Lewis Buxton to fire in his fourth goal in five games.
Buxton had a nightmare afternoon. The young defender, on-loan from Portsmouth, conceded a 24th minute penalty when he brought down Nathan Jones, who was also recalled by McGhee.
Leon Knight converted his 21st goal of the season from the spot as Brighton moved back in to the play-off places.
Buxtons misery was complete when he was substituted early in the second half.
Bournemouth never threatened to launch a come back after the break and they are now six points adrift of Brighton.
McGhee said "It was a good performance. Scoring an early goal helped and the switch back to 4-4-2 worked for us.
I saw Gary Hart and Nathan Jones score for the reserves in the week and they looked determined to win their places back." Bournemouth boss Sean O' Driscoll said "It was a poor game. The ball spent a lot of time out of play and we were punished for some bad mistakes.
We have helped Brighton out by signing their petition for a new stadium and perhaps the deputy Prime Minister John Prescott will give it to them now out of sympathy for the supporters."