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A clinical finish by City old boy Dean Windass had already given Nicky Law's injury ravaged side a mountain to climb, but the unlikely became the impossible when stand-in skipper Gus Uhlenbeek saw red in the 56th minute.
The former Sheffield United man was given a straight red by referee David Pugh, for a lunge on midfielder Michael Tonge and City's patched up side never looked like holding out.
Windass had been the architect of the Blades' opener when his darting run into the box undid the City defence and Kabba was left unmarked inside the six-yard box to side foot home his team mate's cross.
Kabba returned the favour five minutes before the break when he fed the ball out to Windass and the on-loan Middlesborough man fired a low drive past Bradford's new keeper Boaz Myhill.
The England Under-20 shot stopper, who has joined Bradford on loan from Aston Villa, endured a nightmare debut as his under strength team mates were overwhelmed.
City had a lucky escape when Sean Murphy's header was cleared off the line but there were no let-offs after Uhlenbeek's departure.
After 59 minutes Murphy rose highest to meet a free kick and this time Myhill was picking the ball out of the net.
The visitors now had the bit between their teeth and the first win in five games was secured with two more late goals.
After Andy Myers brought down Tonge and midfielder Michael Brown coolly despatched the 79th minute penalty.
And with two minutes left Myhill failed to hold a cross, allowing substitute Carl Asaba to sweep the loose ball into an empty net