The home side took the lead seven minutes into the second half with full-back Michael Duff claiming his first ever goal for the club as his shot was deflected off defender Simon Grand and past keeper Keiren Westwood.
The visitors never gave up and got back on level terms when Peter Murphy's free-kick could only be touched on to the crossbar by goalkeeper Danny Coyne for the rebound to be converted by the head of Adam Murray.
All Carlisle's hard work, however, came to nothing as Burnley's teenage substitute Kyle Lafferty set up striker Akinbiyi for the winner.