Sean Dyche's men made it back-to-back wins as they recovered from an early setback to wrap up their fifth away win of the season and move to within two points of the Cherries.
For Eddie Howe's side it is now just one win from their last nine matches in the Premier League, with their strong start to the campaign keeping them clear of trouble.
With their first attack of the game, Bournemouth broke the deadlock, albeit in fortuitous circumstances. Ryan Fraser’s free-kick was flicked on inside the box and, in an aerial challenge with Nathan Ake, Ashley Barnes headed the ball into his own net.
The Cherries' opener seem to spark Burnley into life and they levelled after 18 minutes, Bournemouth goalkeeper Asmir Begovic getting caught under the ball as Chris Wood stole in at the far post to nod home.
Two minutes later and the Clarets led, Chris Mepham’s clearance falling straight to Ashley Westwood, who picked out the bottom corner from 12 yards out.
An out-of-sorts Callum Wilson then missed a great chance to equalise for Bournemouth, heading over from point-blank range from David Brooks’ cross.
Burnley seized control of the game after the break and wasted two good chances to add to their lead, Ben Mee and Barnes failing to hit the target before the third eventually arrived on 57 minutes.
It was another defensive mix-up that gifted Barnes his ninth Premier League goal of the season, Begovic spilling Charlie Taylor’s cross under pressure from Wood, who teed up Barnes to fire into the roof of the net.
Bournemouth were left aggrieved by referee Martin Atkinson’s decision to let the goal standing, believing Taylor had carried the ball out of play, while Begovic appeared to have been fouled by Wood although replays showed it was an error by Bosnian.
Bee then flashed a header wide as Burnley went searching for a fourth but three was enough, with Bournemouth's miserable day being summed up by Wilson missing a great chance late on.