The visitors had the edge in a low-key first half and Carlisle goalkeeper Keiren Westwood was twice well positioned to clutch a header from teenage striker Sam Vokes and a cross-cum-shot by Max Alain-Gradel.
Alain-Gradel also saw his deflected shot rebound to safety off the post shortly afterwards with Westwood well beaten, while Vokes should have done better with a close-range chance after 28 minutes but completely mis-kicked from six yards.
The Cumbrians were fired up after the break and, after Danny Graham had missed a great headed chance, they took the lead through Scott Dobie in the 57th minute.
Headers from Peter Murphy and Grant Smith put the visitors defence under pressure, and Dobie was on hand a couple of yards out to force the ball past David Forde.
Bournemouth drew level when Lee Bradbury's free-kick on the right was nodded on by Vokes and Brett Pitman was quickest to react, poking the ball past Westwood from eight yards in the 68th minute.