Goals from Darren Bent, Ashkan Dejagah and Steve Sidwell saw Rene Meulensteen's men set up a trip to Sheffield United as Fulham went some way to making amends for their miserable home defeat to Sunderland in the Premier League.
The hosts had survived an early scare when Robert Snodgrass rattled the crossbar and took the lead in the 16th minute with Bent converting from 12 yards out after Pajtim Kasami was released down the right and Javier Garrido failed to deal with his pull-back.
And they doubled their lead in the 41st minute when Mark Bunn's poor clearance gave Sascha Riether the chance to play in Kacaniklic and his ball across the face of goal was converted from close range by Dejagah.
As in the first half, Norwich started the second brightly, but they struggled to create any real opportunities to test returning Cottagers goalkeeper Maarten Stekelenburg before Sidwell made the game safe for the hosts in the 68th minute.
Garrido had made an excellent clearance from under his own bar with Dejagah ready to convert Kacaniklic's cross, but the danger was not over and when Riether swung the ball back in, Sidwell climbed at the far post to convert as Norwich's winless run stretched to eight games.
Stekelenburg was straight into the action from the off, denying Nathan Redmond after Josh Murphy had threaded him clean through on goal.
The Holland keeper could only watch on as Robert Snodgrass cut inside on the edge of the area and launched a curling shot against the crossbar.
Brede Hangeland's slip then allowed Johan Elmander to nip in behind, but his cross eluded a team-mate. The Swede crept his way to another chance moments later, but could only turn a low cross wide of Stekelenburg's right post.
Sidwell fired in Fulham's first shot on the 10-minute mark, but Bunn was alive to the low 20-yard drive.
Snodgrass fed Redmond again, but the former Birmingham man drilled his shot wide from the edge of the area. Josh Murphy skipped clear of Richardson next, only for Burn to clear his near-post cross.
Garrido's foul on Dejagah yielded a 20-yard free kick and a sight of goal for Fulham, but Kasami's effort drew nothing more than a routine stop from Bunn.
So circumspect for the first quarter-hour, suddenly Fulham produced the match's first top-class move, capped with a neat finish from Bent.
Riether sent Kasami down the right flank after a smart flick-off from Dejagah. The Iran midfielder's near-post run and intelligent dummy opened the door for Bent, and the on-loan Aston Villa forward delivered a patient finish.
Criticism of Bent has been commonplace this term, but the 29-year-old began the move that brought Fulham's second goal.
Bent's header capitalised on Bunn's wayward clearance, before Kacaniklic's square ball offered Dejagah a tap-in.
Burn then got just enough to deflect a Norwich cross as far as Riether, who blasted behind for a corner to complete the clearance. Bradley Johnson reached the neat set-piece, but could only nod over the bar.
Hughton sacrificed Redmond's creative talents in favour of a second striker at the break, adding Ricky van Wolfswinkel to the fray.
Fulham's desperation not to concede sucked them ever deeper into their own half as they allowed Norwich to dominate territory.
Captain Sebastien Bassong back-pedalled himself free at the far post from a corner after Sidwell had conceded a needless free kick, But the former Newcastle defender could not generate any power.
Fulham then wrapped up safe passage to the fourth round, thanks to Sidwell's powerful far-post header.
Garrido denied Dejagah with a last-ditch headed clearance, but his despairing interjection only gifted Riether time and space to cross from the right. The German full-back took his time, picked his target, and Sidwell did the rest.