Substitute McCann fired home in the 105th minute from a Brian Easton cross to save the Championship side's blushes after the League Two team's dramatic late revival had forced the extra half-hour on Tuesday night.
McCann had been dropped to the bench by boss Eddie Howe and the hosts started brightly with Kieran Trippier dragging a shot just wide before Jason Price missed a chance for Barnet.
Jay Rodriguez opened the scoring for Burnley in the 38th minute when he deflected home a Trippier cross but the visitors were undaunted and Sam Deering responded by crashing a shot over the bar.
Clever work in the box by Wade Elliott resulted in the striker firing Burnley 2-0 ahead in the 64th minute and it seemed like the visitors' brave resistance had finally crumbled.
But Barnet hit back with veteran striker Steve Kabba, on as a 73rd-minute replacement for Price, sliding in at the far post to reduce the deficit from a Mark Marshall cross.
And, after Rodriguez had missed for the Clarets from close range, Barnet forced extra time in dramatic fashion when another substitute, Ricky Holmes, prodded home the equaliser in the fourth minute of stoppage time.
Burnley finally settled in the extra period with McCann - on at the end of normal time for Keith Treacy - nudging them back in front before Trippier, Zavon Hines and Ross Wallace all missed chances to make it four.