The home side secured their first league victory of the season last weekend at Hartlepool United, and they had the best possible start when Tom Miller met Danny Grainger's free kick to head home his second goal in as many League Two games.
Within 90 seconds of that goal the Bees had wiped out the lead altogether when Gambin crossed for Michael Gash who this time made no mistake to beat Mark Gillespie, and they could have completed a remarkable turnaround before the break had John Akinde been able to find the net.
Within five minutes of the restart, Akinde had a goal correctly ruled out for offside and missed a golden opportunity after robbing Michael Raynes, and Gash was denied a second goal by a deflection as Barnet dominated the second half.
However, it was Carlisle who retook the lead against the run of play when Alexander McQueen's low cross was tapped home by Asamoah to give the home side something to hold on to for the remaining 25 minutes.
Dembele headed against the bar for a second time and Gillespie saved from Curtis Weston late on as Barnet pushed for a point, but the home side held out for a second victory of the season.