First-half goals from Wes Fogden and Eunan O'Kane set up the win for the Cherries and although Mark Beck's header gave Carlisle some hope, Marc Pugh clinched it for the south coast men in time added on.
A string of early chances fell to Carlisle winger Matty Robson, who was twice denied by Cherries goalkeeper Shwan Jalal before blasting wide from well inside the area when he should have found the target.
Bournemouth scored with their first real shot at goal on the half-hour. Carlisle failed to clear Pugh's corner from the left and Fogden drove the ball home through a crowd of players from the edge of the area.
Cherries doubled their lead within the space of six minutes when clever footwork by Pugh allowed Fogden to set up a chance for O'Kane, who hammered a rising shot past Carlisle goalkeeper Adam Collin from 10 yards.
A drab second half, largely dominated by Bournemouth, sparked back to life in the 72nd minute when Jalal missed Jon-Paul McGovern's free-kick and Beck got up at the far post to nod home his third goal in two games before Jamal made amends with a save which kept out Beck's low drive before the Cherries had the last word when Simon Francis cut the ball back from the by-line for Pugh to side-foot home from six yards.