Tony Naylor was the toast of the town with two opportunist first-half goals and Hugh McAuley completed the scoring six minutes from time.
Naylor's first goal on the half-hour saw him intercept Malky Mackay's headed backpass before rounding the stranded City keeper and finding the net with a low right-foot angled drive.
Two minutes after this Mark Yates was booked for a foul on David Nielsen.
The Town keeper Steve Book did well to nick the ball off Nielsen's toe when he was about to convert an Iwan Roberts pass.
Roberts headed over when he should have found the target and was then booked for a foul on Yates two minutes before half time.
There was further embarrassment for the home side on the stroke of half time when Naylor beat Darren Kenton to Michael Duff's throughball and notched his second goal from 18 yards.
Cheltenham's grip on the game was underlined by the fact that it took Norwich 55 minutes to get a shot on target.
City boss Nigel Worthington used all three of his substitutes but it was to no avail and McAuley killed off the game in the 83rd minute when he rifled in an unstoppable shot from Russell Milton's indirect free kick after Green had been adjudged to have handled a backpass.
The keeper claimed Mackay's back pass had taken a deflection off a Town player.
However, no-one would deny that second division Cheltenham were worthy winners on the night and justifiably make progress into the second round of the cup for the first time since their promotion to the League four years ago