The game was deep into injury time when Puncheon drilled home a free-kick to notch up his fourth goal in four games and seal the points for Barnet in fortuitous style after a dominant second-half showing from Stuart McCall's visitors.
The late drama came after Bradford midfielder Eddie Johnson had atoned for a 42nd-minute own goal with a stinging 20-yard equaliser just after the break.
After bagging their first home win in eight months against Wrexham last weekend, McCall's men had made the journey south in confident mood.
But City have had little joy on the road in recent months, and suffered an early blow when midfielder Paul Evans came off worse following a clash of heads with Puncheon.
Evans was replaced by Scott Phelan, and the teenager quickly adapted to the pace of the game as City lost no time in carrying the fight to their hosts. Winger Omar Daley tested Lee Harrison with a stinging long-range shot before Dubliner Barry Conlon forced a smothering save from the Barnet stopper.
Up to that point, however, it was Barnet who had looked the likelier to score. With the lively Ashley Carew a constant source of threat down the right wing, it was only some tame finishing from front man Adam Birchall that prevented the Bees from establishing an early lead.
But for all Barnet's attacking impetus, it took a stroke of good fortune to give the home side the lead. With half-time approaching, Puncheon's left-footed free-kick from the right bamboozled Johnson, who inadvertently fired the ball past Donovan Ricketts and into his own net.
But as Bradford emerged from the break with all guns blazing, it took Johnson just seven minutes to atone for his error, the former Manchester United man latching on to a loose ball to fire an unstoppable shot past Harrison from 20 yards.
Seconds later Barnet front man Anthony Thomas almost responded in spectacular style, but Ricketts was equal to the challenge and City were soon pouring forward in search of a winner.
Daley can count himself unlucky to have missed with a trio of chances, first when he stumbled inside the six-yard box as he set himself to shoot, and then with two flashing long-range strikes, both of which brushed the outside of Harrison's post before trickling to safety.
And with seconds remaining Puncheon made Daley pay dearly for his profligacy with a goal that sent the Underhill into delirium.