Darren Bent opened the scoring for Fulham in the 16th minute, driving home from a Patjim Kasami cross, and Ashkan Dejagah added a second before the break.
Steve Sidwell made sure of progression in the 68th minute, as Fulham's unbeaten run against Norwich stretched to a remarkable 15 matches in all competitions.
Fulham will be wary of an upset in round four though, as League One outfit Sheffield United have already knocked Premier League opposition out of the competition with their shock win at Aston Villa.
Rene Meulensteen's mind appeared focused on Premier League survival as he he made nine changes from Fulham's 4-1 top-flight defeat to Sunderland on Saturday, including a return for Brede Hangeland, who had been out since October following back surgery.
Chris Hughton, whose side are also battling against the drop, made six alterations to the side that lost 2-0 at Everton at the weekend.
Norwich gave their hosts a scare in the opening minutes at a sparsely populated Craven Cottage - Nathan Redmond drawing a save from Maarten Stekelenburg before Robert Snodgrass struck the crossbar with a superb curling effort from 20 yards.
Following the early threat from the visitors, Sidwell had Fulham's first meaningful shot at goal when he fired straight at Mark Bunn from the edge of the area in the 10th minute.
And it was Meulensteen's men who broke the deadlock six minutes later, when a Patjim Kasami pull-back found Bent in space in the centre of the box, and the striker made no mistake in drilling the ball beyond Bunn.
Fulham continued to press, and Sebastien Bassong had to be strong to outmuscle Bent and make a vital clearance from an Alexander Kacaniklic cross in the 27th minute.
The hosts’ persistent pressure paid off four minutes before the break, as Kacaniklic broke down the right and squared the ball to to the edge of the six-yard box, teeing up a simple tap in for Dejagah.
Norwich started the second half strongly, but chances were few and far between until Bassong got his head to a Snodgrass free-kick just after the hour mark, with Stekelenburg equal to the effort.
However, Sidwell put the result beyond doubt moments later as he powerfully headed home a Sascha Riether cross from the right.
Norwich looked a beaten side thereafter, and failed to pose too much of a threat to Stekelenburg's goal.
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