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 then made sure of progression in the 68th minute as the hosts' unbeaten run against the Canaries stretched to a remarkable 15 matches in all competitions.
Fulham will be wary of an upset in round four, though, as League One side Sheffield United have already knocked Premier League opposition out of the competition with their shock win at Aston Villa.
|VIEW FROM CRAVEN COTTAGE
|By Liam Twomey
Rene Meulensteen voiced his intention to take this match seriously in his pre-game notes and he will be pleased with the reaction that he got from his Fulham players after a shattering home loss to Sunderland.
Then, as here, Steve Sidwell was at the heart of everything that the home side did well - the difference was that those around him did their jobs with distinction too. Ashkan Dejagah and Alexander Kacaniklic were dangerous, Darren Bent scored and Dan Burn enjoyed a drama-free senior debut alongside returning veteran Brede Hangeland.
Norwich City offered very little after a bright start which saw Maarten Stekelenburg do well to deny Nathan Redmond and Robert Snodgrass find the crossbar with a superb 25-yard curler.
The FA Cup is not a priority for Chris Hughton but this result offers no respite to the pressure that he is under and the outcome of Saturday's Premier League clash with Hull City may now prove decisive.
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, though, 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.