Two goals in six second half minutes gave the home side the advantage with the Canaries grabbing a consolation at the death.
In chilly conditions at Brockhall it was an even first half, Jordan Preston going close for Rovers on 12 minutes, firing just wide after a lofted pass from Bradley Bauress.
Norwich responded by rattling the crossbar on 24, David Raya couldn't claim a corner kick and as the ball deflected high Reece Johnson-Hall headed onto the woodwork and over.
Three minutes later and it was Rovers' turn to hit the frame of the goal in a trio of good opportunities. Jordan Preston was first denied by a smart save and as the resulting corner was cleared Tom Brown smacked his long-range strike against the crossbar.
The chances weren't over as the ball broke to Tom Paul in the box, the defender unfortunate to see his first-time shot superbly saved.
After a tight first 45, Rovers were fast out of the blocks at the start of the second with a goal inside a minute.
A cross from Dean Rittenberg saw Tom Brown fire goalwards with Jordan Preston getting the final touch as the ball deflected in although he didn't know too much about it.
Six minutes later and Rovers' lead was doubled, a swift attack down the left from Modou Cham saw the young attacker find Callum Harris with his cross and he fired home first-time for his second of the campaign.
It was almost 3-0 moments later but Jordan Preston was brilliantly denied by the Canaries' 'keeper.
The closing stages saw Norwich look for a way back into the game, the young Rovers defence standing up well to a spell of pressure. Sam Lavelle diverting one effort wide of goal before central defensive partner Tom Paul made an important blocked in the area.
The home side's copybook was blotted however when deep into injury time goalkeeper David Raya sliced his clearance straight to Norwich striker Reece Johnson-Hall and he steered the ball into the empty net.
But it proved to be the last action of the game as the Under-18s registered their third win over Norwich City this campaign.
"With Academy football you never know what you will get but I was delighted with the response from the lads after the weekend," said Terry McPhillips.
"We've a good group and they worked hard to get the win in a tight competitive match."
ROVERS: Raya, Anderson, Bauress, Brown, Lavelle, Paul, Rittenberg, Torres (sub Wassi 67 mins), Preston, Harris, Cham.
Subs not used: Pero, Doyle, Butterwick, Wall.