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.
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 looked a beaten side thereafter, and failed to pose too much of a threat to Stekelenburg's goal.
