Goals from Iwan Roberts - with a twice taken penalty - and Phil Mulryne, in the 90th minute, securing the points after they played the whole of the second half with ten men following the dismissal of Kevin Harper on the stroke of half time.
Derby kicked off attacking the River End goal and enjoyed the best of the opening exchanges. The visitors were inches away from scoring when Daniele Dichio's fierce drive hit Robert Green's left-hand post.
In the 20th minute Peter Crouch wasted City's best chance when his chip from a Gary Holt pass cleared the bar.
Paul McVeigh brought a good save from Lee Grant with a crisp snap shot after 27 minutes.
Five minutes later Green did well to hold a rising drive from Lee Morris.
When play switched to the other end Adam Drury went down in the penalty area under pressure from Dave Walton but the referee infuriated the home fans by waving play on.
Five minutes before half time Harper ran through the Derby defence but curled his shot wide.
Francis did better with a long-range effort but Grant parried his shot and grabbed it at the second attempt.
In first-half stoppage time Harper was sent off for a dangerous tackle on Luciano Zavagno.
Norwich started the second half in determined mood, playing better with ten men than they had with 11.
The referee further frustrated the Norwich fans when he awarded a penalty for what appeared to be a dive by Zavagno after a challenge by McVeigh.
Ian Taylor converted the spot kick his right foot drive going straight down the middle and into the roof of the net.
Green kept Norwich in the game with an acrobatic save low to his left to keep out a header from Dichio.
With a quarter of an hour to play Nigel Worthington made a triple substitution which turned the game around with Roberts, Mulryne and Ryan Jarvis replacing Crouch, McVeigh and Mark Rivers.
Jarvis brought a reflex save from Grant with his first touch before the other two substitutes rescued the Canaries.
Roberts beat Grant with a twice taken penalty after the Rams keeper had moved off his line.
Green pulled off another classy save to deny Taylor and in the 90th minute Mulryne collected a pass from Marc Edworthy and beat Grant with a low right-foot shot inside the keeper's right-hand post.
Deep into stoppage time Zavagno palmed the ball past Green and was booked for his trouble as the home fans went home happy.