The visitors were quick out of the starting blocks, winning three corners in quick succession early on which culminated in Chris Wood smashing a header off the bar and out of play.
The former Leeds United man continued to threaten and was rewarded with the game's opening goal on 36 minutes, poking home from six yards after Robbie Brady's initial effort was deflected by Steve Cook.
Burnley doubled their lead in the 65th minute when a counter-attack led by Johann Berg Gudmundsson saw the ball worked to Brady, who shifted the ball onto his weaker right foot and curled the ball into the top right-hand corner of the net.
With 11 minutes left to play, Bournemouth set up a nervy finish as Ryan Fraser's cross from the right fell to Jermain Defoe, who skewed the ball across the face of the box where Joshua King was on hand to bundle home.
The Cherries piled men forward in search of an equaliser but were unable to find another goal as the Clarets continued their impressive form in the league, while Bournemouth remain 15th in the standings.
Bournemouth (4-4-1-1): Begovic; Francis, Cook, Ake, Daniels; Ibe (Fraser 69'), Surman, Arter (Cook 58'), Pugh (Defoe 64'); King; Wilson
Burnley (4-4-1-1): Pope; Bardsley, Tarkowski, Mee, Ward; Gudmundsson, Defour (Westwood 75'), Cork, Brady; Hendrick (Barnes 86'); Wood (Vokes 90')