The midfielder fired in off the bar within seconds of coming on after latching onto Adam Armstrong's header.
Bradford, whose outside hopes of automatic League One promotion have now all but gone, went close through Josh Cullen and Jamie Proctor.
Their miserable night was completed when substitute Steven Davies was sent off for an off-the-ball incident.
The Bantams have dropped to fifth but stay in the play-off spots, while Coventry are 11th, six points shy of the top six, after their second win in four days.
Sky Blues boss Tony Mowbray told BBC Coventry & Warwickshire:
"Andy Rose is a good player. He has a great future at this club. I had a long chat with him before the game, as he was the only change from the weekend.
"He is a fantastic athlete. He is 6ft 2ins, he can head it, he can run, he can spray long passes. His time is going to come.
"The goals had dried up and we were losing matches, but I have always been proud of my players. I have never had to question their desire."
Substitute Andy Rose scored with his first touch as Coventry City edged to victory over 10-man Bradford City.