Boss Sean Dyche made eight changes to the side that picked up a point at home to Brighton in the Premier League, with Anthony Glennon a late entry into the starting line-up following a clerical oversight that impacted on Erik Pieters' involvement.
Sam Surridge put the Cherries ahead midway through the first half when tapping home Jack Stacey's centre from close range.
Former Burnley winger Junior Stanislas then wrapped things up from the spot in the 88th minute with an emphatic finish past Bailey Peacock-Farrell after Kevin Long had committed a foul.
Here are the RATINGS.
1. Bailey Peacock-Farrell 4.8
Didn't have a great deal to do, other than clawing Stephens' wayward header off the line. Left exposed for the opening goal and faced a number of shots, with the majority off target.
2. Phil Bardsley 4.7
Caught far too narrow at times and Rico exploited the space down the side. Wasn't given an easy ride by Stanislas and frustration kicked in late on when he picked up a booking for a rash challenge. Went close with a left-footed effort in the second half.
3. Anthony Glennon 4.1
His introduction at the last minute won't have helped following a clerical oversight, but he was given a torrid time. Brooks had the freedom of the flank and Stacey was a threat on the overlap. Opening goal came from his side.
4. Kevin Long 4.2
Wasn't quick enough to react when Surridge stepped up to convert Stacey's cross. Caught out for pace on occasions and conceded the penalty that gifted the Cherries the tie.