Bournemouth finish in 12th position in the league table, while Burnley had already secured seventh place and a spot in next season's Europa League.
The opening goal of the game went the way of the home side as Ashley Westwood's strike took a deflection off Wood and headed into the back of the net.
Bournemouth were the better side after the restart with Jordon Ibe flashing a shot across goal, and the equaliser finally came during the final quarter, with King applying the finishing touch after latching on a pass from Daniels.
As full time edged nearer, Ibe and Wilson both sent long-range attempts off target but Bournemouth got the winner that their play deserved in the third minute of added-on time.
Substitute Jermain Defoe broke clear and after pulling the ball back for Wilson, the forward did the rest to give the south coast club back-to-back wins to end the season.
BURNLEY (4-5-1): Pope; Lowton, Tarkowski, Long, Ward; Westwood, Cork, Lennon (McNeil 94'), Hendrick, Gudmundsson (Wells 80'); Wood (Vokes 61')
BOURNEMOUTH (4-4-1-1): Begovic; Daniels, S.Cook, Ake, Mings (Defoe 66'); Fraser, Surman, Hyndman (Gosling 59'), Ibe; Mousset (Wilson 59'); King