Sam Surridge put the Championship club ahead after connecting with Jack Stacey’s low cross.
Junior Stanislas added the second from the penalty spot after Surridge was fouled.
Bournemouth, who have won both of their games under caretaker-manager Jonathan Woodgate, were deserved winners.
On a bad night for the Clarets, they were forced to make a late change when it was pointed out that defender Erik Pieters, who was named in the starting line-up, had been booked in the previous two rounds and was therefore suspended.
Bournemouth keeper Asmir Begovic saved from Matej Vydra and Dwight McNeil, while Jay Rodriguez blazed a great chance over from close range.
But the Cherries, who sacked Jason Tindall last week, were well organised and are in the quarter-finals for only the second time in their history.
Former Real Madrid and England defender Woodgate was coy when asked after the tie about the possibility of him landing the job full-time
Let’s not get carried away, I’ve got Nottingham Forest for the next match. I know football can come back and bite you, he said.
The visitors made the breakthrough against a Burnley side showing eight changes after David Brooks found Stacey, who crossed for Surridge to score from inside the six-yard area.
Burnley keeper Bailey Peacock-Farrell saved a header from team-mate Dale Stephens which was heading for the net.
Stanislav’s late penalty ended any hope of a Burnley comeback and Bournemouth find themselves one win away from a semi-final at Wembley.
The draw for the quarter-finals is on Thursday, with matches to take place across the weekend of 20-21 March.