Matty Pattison was the provider for goals scored by Luke Chadwick and Jamie Cureton.
The visitors kicked off attacking the River End goal and soon showed why they have had such a good away record.
They looked solid in defence and dangerous on the break with Dele Adebola causing problems with his physical presence.
Norwich did have the ball in the net after 26 minutes but an offside flag cut short the celebrations as the ball entered the net via Dion Dublin's back.
However, Norwich did take the lead in the 34th minute and it was Pattison's low cross from the left, touched on by Dublin to Chadwick, and his first-time left-foot shot beat the keeper low to his right from 12 yards.
David Marshall saved Adebola's header on the stroke of half-time to ensure that Norwich went in with a rare first-half lead.
Eight minutes after the break, Kevin Kyle limped off to be replaced by Ellery Cairo.
Ben Turner got away with a blatant hand ball when he blocked Jon Otsemobor's cross with Andy D'Urso failing to spot the offence and award a penalty.
That seemed to raise the tempo of the game and both Chadwick and Cureton had goal-bound efforts blocked inside the Coventry penalty area.
Wayne Andrews came on from Stephen Hughes and went close to scoring with a glancing header.
At the other end, Cureton fired wide before Chadwick made way for Lee Croft.
Dublin's header flashed wide, but a second goal came for Norwich in the 77th minute.
Croft found Pattison on the left and his low cross was blasted home by Cureton from close range at the far post.
Marshall then did well to keep out Andrews' shot and Dimitrios Konstantopoulos saved from Cureton.
As the visitors forced a string of corners, Marshall saved Cairo's header and an angled drive from late substitute Julian Gray, ensuring that at long last the Canary fans were able to celebrate.