Lee Trundle kept Swansea on top of the league with his ninth goal this season.
The striker was in the right place on the half hour mark to fire home after Bournemouth goalkeeper Gareth Stewart had palmed away Izzy Iriekpen's shot.
Swansea wasted a host of chances to double their lead and they needed a fine Willy Gueret save to deny James Keene's effort before the break.
Adebayo Akinfenwa then hit the bar for the visitors before Gueret tipped James Hayter's header round the post.
Bournemouth: Stewart, Hart, Gowling, Young, O'Connor, Foley, Stock, Browning, Surman, Hayter, Keene.
Subs Not Used: Rodrigues, Pitman, Cooper, Cooke, Coutts.
Swansea: Gueret, Tate, Iriekpen, Austin, Ricketts, Britton, Bean, Martinez, McLeod, Akinfenwa (Robinson 84), Trundle.
Subs Not Used: Murphy, O'Leary, Monk, Goodfellow.
Booked: Bean.
Goals: Trundle 30.
Att: 5,750
Ref: G Hegley (Hertfordshire).