After a first half in which Rovers dominated yet somehow trailed to Jamar Loza's free header, the visitors upped the ante in the second period and ran out comfortable winners.
Loza, completed his hat-trick with two further goals before Josh Murphy grabbed a late brace to give the scoreline a one-side look.
Had Rovers' taken their chances in the first half when they had the better of the play the game could have oh-so different.
David Dunn came through 69 minutes, whilst fellow seniors David Goodwillie, Josh Morris and Jake Kean all played the full match.
Rovers had named a stronger starting XI than usual for the clash with the Canaries as David Dunn, David Goodwillie, Josh Morris and Jake Kean were all involved.
Only a fingertip save from Carlo Nash prevented Rovers from taking an eleventh minute lead as play was switched from left to right to where Connor Mahoney took aim. His left-footed strike seemed destined for the top corner until the well-travelled goalkeeper's timely intervention.
Opportunities on the break seemed to the the Canaries' best option, with wide twins Josh and Jacob Murphy seeing plenty of the ball.
On 19 minutes space opened up for Jacob Murphy to try his luck from 25 yards but his shot wasn't troubling Jake Kean's goal as it flew high over the bar.
At the other end David Goodwillie fired goalwards from a similar distance, but the Scotman's shot was also off target.
Then Rovers defender Sam Lavelle stepped up to rob a Norwich attacker on half way, David Dunn took over to find Goodwillie but Nash saved his shot well at his near post.
Rovers were seeing plenty of the ball, David Dunn playing a key role in keeping possession and bringing fellow forwards David Goodwillie and Connor Mahoney into play.
One such move saw Dunn again find Goodwillie, but Nash again was equal to his efforts. Play was immediately switched downfield for Jacob Murphy he he lashed over the bar.
But for all of Rovers' good play it was the visitors who took the lead nine minutes before the break.
A corner from the right by Josh Murphy found striker Jamar Loza totally unmarked on the penalty spot and he had the simplest of tasks to head home.
It was a very disappointing goal to concede.
David Dunn was inches over with a free-kick before Jake Kean made a fine save to deny Jacob Murphy in the closing stages of the half.
Neither side made a change at the break but it was the visitors who reappeared with a much greater attacking threat, Jamar Loza immediately forcing Jake Kean into a save before Josh Murphy fired well over.
The Canaries continued to press and eventually grabbed the all-important second goal when Jacob Murphy's ball in from the right found Loza and he took a touch inside onto his left foot before firing an unstoppable shot into the top corner.
David Dunn made way for Dean Rittenberg soon after, getting a useful 69 minutes under his belt, but the young Rovers struggled to regain any foothold in the game.
A third goal soon followed and it was a third for Loza on 76 minutes. A lovely turn in the Rovers half saw him find Carlton Morris on the right, the ball was centred with Loza somehow applying the finishing touch to poke the ball home.
With Rovers down to ten men whilst after Hugo Fernandez was instructed to put on some shin pads, Norwich took full advantage. This time Loza turning provider in cutting the ball back for Josh Murphy strike beyond Jake Kean.
Salt was then well and truly rubbed into the Rovers' wounds as a fifth was added in the 92 minute. With almost the last kick of the game Josh Murphy grabbed his second of the game from close-range.
It was a disappointing end to what had started off as a positive night.
ROVERS: Kean, Fernandez, Morris, Pierce, Lavelle, Tumwa, Mahoney, Dunn (sub Rittenberg 69 mins), Goodwillie, Payne (c) (sub Beesley 62 mins), Bauress (sub Preston 81 mins).
Subs not used: Urwin, Pero.
NORWICH: Nash, Barker, Toffolo, Hall-Johnson, McFadden, Gafaiti (sub Lokko 85 mins), Jacob Murphy, Hood (sub Norman 82 mins), Morris (sub Young 78 mins), Jamar Loza, Josh Murphy.
Subs not used: Killip.
Goals: Loza 36, 66, 76 mins, Josh Murphy 82, 90+2 mins.