They had to come from behind with two goals in the second half after the home side had been reduced to ten men on the half-hour mark.
Cheltenham captain John Finnigan was given his marching orders following an altercation with veteran striker Dean Windass and that proved the turning point of the game.
Cheltenham took the lead in the fourth minute when a good build-up involving Craig Armstrong, Steve Guinan and Brian Wilson ended with a Wilson cross from the left.
Bradford defender Richard Edghill made a mess of his clearing header sending it straight to Kayode Odejayi who made no mistake with a shot from eight yards.
Bradford stormed back with Windass shooting wide from the edge of the box and Cheltenham's stand-in goalkeeper Scott Brown making a reflex save from a towering David Wetherall header.
The controversy began in the 31st minute when Windass went to ground holding his face following an altercation with Finnigan.
Referee Clive Penton allowed play to continue but at the next stoppage he held a long conversation with the fourth official, following which Finnigan was sent off and Windass booked.
After being reduced to ten men, Cheltenham replaced striker Guinan with midfielder David Bird and sat back behind the ball, using the speedy Odejayi as a lone front runner.
They could only hold out until the 52nd minute when David Graham met a Lee Holmes cross with a stooping header to level the scores.
Then, with 21 minutes left, Bradford sealed the points when the impressive Jermaine Johnson collected a pass from Windass and surged into the box before beating Brown with a low finish from a tight angle.
Bradford created a series of other chances in the second half as they made the extra man count to good effect, although Cheltenham almost levelled 12 minutes from time when Odejayi took a pass from Armstrong on his chest and got a clear sight of goal but fired his shot over the bar.