The Canaries took a 32nd-minute lead in the League One clash through Chris Martin's low shot after a mistake by Rovers defender Daniel Jones.
It was 2-0 on the stroke of half-time when Anthony McNamee set up substitute Oli Johnson to score with his first touch.
Midfielder Stephen Hughes ended any doubt about the outcome by shooting home from close range after Martin had set him up in the 67th minute.
Rovers striker Daryl Duffy headed against the crossbar from a Ben Swallow cross 10 minutes from time, but the below strength home side were well beaten.
McNamee ran the game from a position just behind the Norwich strikers and nearly 1,800 travelling fans spent most of the game partying.