Rovers were defeated in a 3-1 loss at The Mem, which sees us exit the FA Cup.
Luke McCormick fired the Gas into a deserved first-half lead following early pressure, with a brilliantly taken volley in the 20th-minute. It was, however, a different story in the second-half as goals from Gabriel Sara, Anthony Idah, and Kenny McLean sent the Championship side into the next round.
A trip to Anfield was at stake in the next round of the cup, and the Gas started much the better side at a sell-out Memorial Stadium.
Rovers pressed high and won the ball back in Norwich territory but Chris Martin’s lobbed effort landed just wide of the far post.
Sam Finley’s pinpoint cross-field ball found an unmarked Luke Thomas but Norwich goalkeeper George Long backtracked smartly to stop.
Harvey Vale was next to test Long as his long-range effort was tipped wide of the post.
From the resulting corner, Norwich could only clear to the edge of the penalty area where McCormick fired home a crisp volley into the bottom corner to send the hosts into the lead on 20 minutes.
Soon after, central defender Connor Taylor produced a vital block as Onel Hernandez delivered a fine low ball across the six-yard box.
Matt Cox then helped Sam McCallum’s cross-turned-shot over the crossbar but the Gas kept Norwich at arm's length for the remainder of the half.
Luke Thomas had a chance to make it two on the brink of half-time but his low curling effort drifted just past the far post.
Thomas was in the action again soon after the break, as he shifted the ball to make a yard of space, striking low but Long parried to safety.
Taylor’s anticipation was on show again as his sliding challenge denied a through ball to Idah.
Norwich equalised in the 53rd-minute when McCallum’s fine effort cannoned back off the post, with Sara on hand to tap home.
Just four minutes later, the Canaries broke at pace before Jonathan Rowe was adjudged to have been tripped in the penalty area. Despite Cox’s best efforts to save, Idah’s powerfully struck spot-kick gave Norwich the lead just before the hour mark.
Just minutes later, Cox did well firstly to deny Idah after he was played through one-on-one, and then to keep out Sara’s low-drilled effort.
Evans played an inch-perfect forward pass for Aaron Collins who powered towards goal but Long dealt with the danger.
The Gas produced a fine-flowing move as Thomas found Chris Martin at the back post, but he was spectacularly denied by an onrushing Long, as Rovers looked likely to find an equaliser.
Evans was next to try his luck as the hosts pressed but his effort from distance landed narrowly wide of the far corner.
City threatened to grab another when Jack Stacey’s delicate cross found substitute Ashley Barnes, but he failed to guide his effort goalwards.
Norwich did made it three after 86 minutes, catching Rovers on the counter, when McLean fired home a left-footed strike from distance that left Cox with no chance.
Next up for Rovers is this weekend's Sky Bet League One fixture against Blackpool at The Mem, on Saturday 20th March (3pm KO).
Bristol Rovers: Cox, Grant, Crama, Taylor, Thomas, Finley (Marquis 79'), Evans, Vale, McCormick (68'), Martin, Collins (79')
Subs: Sinclair (79'), Marquis (79'), Hunt (68') Gordon, Brown, Gibbons, Hoole, Hall, Shaw
Norwich City: Long, Stacey (90+3'), Batth (27'), Gibson, McCallum, Sorensen, McLean, Hernandez (Sainz 64'), Sara, Rowe, Idah (64')
Subs: Fisher (90+3'), Nunez (27'), Barnes (64'), Gibbs, Fassnacht, Placheta, Gunn, Ansen
Referee: Andy Davies
Attendance: 10,335 (828 away)