Former Burnley forward Junior Stanislas opened the scoring between the two sides after latching on to a well weighted ball from Lewis Cook.
Sprinting clear of the defence, he was able to slot calmly into the bottom corner from 18 yards out.
Sam Vokes equalised for the away side with a powerful headed goal before Josh King restored the lead for Bournemouth just two minutes later, converting a cross from Ryan Fraser. The result leaves Burnley 14th in the table, six clear of Hull in the Premier League relegation zone.
Player Ratings
Tom Heaton - 5 - Caught out by Junior Stanislas for the opening goal of the game and was against found wanting as Josh King put the ball through his legs for the winner. Not a good afternoon for the goalkeeper.
Matt Lowton - 5 - Often the spark for Burnley down the flank, he failed to impress at the weekend. Defensively, he was at fault for Bournemouth's winning goal, with Fraser getting the better of him.
Kevin Long - 6 - Made a good interception early on in the game but struggled on the whole. Another hard afternoon for the centre back.
James Tarkowski - 5 - Again, part blame can be attached to him for the winning goal. King got the wrong side of the centre-back which proved a costly mistake in the outcome of the match.
Stephen Ward - 5 - Like his defensive teammates, it wasn't a great afternoon for the full-back. Stanislas got the better of him on countless opportunities, with his pace proving hard to deal with.
George Boyd - 5 - Little impact on the game for the away side. Brightest moment came when he produced an early shot that fell just wide of the post.
Ashley Westwood - 6 - Typified Burnley's performance on the whole with a sub-par display. Not up to his usual standard in the middle of the park.
Jeff Hendrick - 7 - Arguably the best player on the day and tried to drive his team toward an equaliser with some good and bright distribution. Got forward in support of the attack well with incisive runs.
Scott Arfield - 6 - Got into some bright positions during the match but couldn't make them count for his side.
Ashley Barnes - 5 - Did little in the opening forty five and was subsequently removed from the action. Lacked the pace to provide a real threat on the attack. It just wasn't his game, much like many of his teammates.
Sam Vokes - 7 - Produced a brilliant header for Burnley's goal and was certainly their best and most dangerous player on the day. Dealt well with the aerial battle between two powerful centre-backs and looked sharp.