Ricardo Santos's header gave the Bees the lead, but Akinde's penalty miss was followed by three Carlisle strikes to give the Cumbrian's the victory.
Barnet almost made the perfect start, but Jordan Nicholson's 2nd-minute effort bobbled narrowly wide of the post.
On 15 minutes, John Akinde shrugged off his marker but his effort from 25 yards struck the side netting.
Craig Ross produced a superb diving save away to his left to palm Joyce's goalbound effort around the post.
Shortly after the break, Barnet took a deserved lead. Ryan Watson's corner was powerfully headed home by Ricardo Santos.
On the hour mark, the Bees were handed the perfect opportunity to double their lead as Akinde's heels were clipped as he raced into the area.
Akinde stepped up to take the spot kick but fired it high over the crossbar.
After this miss, the momentum swung Carlisle's way. The Cumbrian's made a triple substitution and one of the replacements, Ashley Nadesan, tied the game with 20 minutes left to play.
Craig Ross made the decision to come out of goal but was beaten by Nadesan who looped the ball over the stranded Barnet 'keeper.
In the 82nd minute, another of the triple change, Richie Bennett, gave the visitors the lead and Nadesan grabbed a third, flicking on Jamie Devitt's free kick.
Barnet - Ross, Clough, Santos, Watson, Weston (Bover '83), Akinde, Tutonda, Vilhete (J. Taylor '77), Nicholls, Sweeney, Nicholson (Coulthirst '77)
Unused - Legg, Shomotun, Sule, Plavotic
Attendance - 1,151 (207 Carlisle)