The promotion-chasing Foxes showed they were more than a match for Premier League opposition, although considering the Cottagers' struggles in the top flight it was difficult to describe the result as a shock.
Just eight league places separate the two sides with Leicester second in the Championship, and they showed the character to bounce back from an 85th-minute equaliser - having led 3-1 - to snatch a 4-3 victory.
Hugo Rodallega's second goal of the season two months after his first, also in this competition, had put the visitors ahead only for the home side to strike twice late on in the first half.
Fulham contributed significantly to their own downfall as individual errors led to both goals.
Less than five minutes later Philippe Senderos, who was the only outfield survivor from Saturday's 2-0 defeat at Southampton, hacked a clearance into Paul Konchesky and Chris Wood gratefully accepted the rebound.
Leicester made it three goals in 12 minutes either side of half-time when Ignasi Miquel headed in only for Rodallega to immediately go down the other end and pull one back.
When Giorgos Karagounis smashed home a free-kick with five minutes remaining Leicester should have been crushed, but they picked themselves up and Dyer delivered the knock-out blow.
The home side's promising start was halted midway through the first half when John Arne Riise's low cross was dummied by Adel Taarabt for the unnarked Rodallega to convert at the far post.
They should have capitalised on that but the Cottagers are lacking in confidence and their opposition were quickly back at them with Konchesky letting one fly from distance before Stekelenburg got down well to hold Wood's low drive.
Stekelenburg was not so assured with his handling four minutes before the break as, when Knockaert drilled in a 30-yard free-kick, he fumbled and Morgan weighed in with his first goal since August 2012.
More calamity was to follow for Fulham as Senderos' clearance rebounded off Konchesky into the path of Wood who comfortably beat the goalkeeper.
Any hope Fulham had of rallying after the break was quickly snuffed out as Leicester leapt on them straight from the restart.
Dyer, on for Konchesky at the interval, blazed wide with a clear sight of goal before, in the 51st minute, Danny Drinkwater's cross was headed in by Miquel.
That appeared to have clinched the win as Fulham had shown little appetite for a fight but the visitors went straight down the other end and Rodallega nodded in Karagounis' cross at the far post.
Suddenly the game became a proper cup tie with Wood's acrobatic volley tipped away by Stekelenburg and Jeffrey Schlupp running right through the middle of Fulham only to be stopped by Aaron Hughes' well-timed block.
Cottagers' substitute Bryan Ruiz shot wastefully straight at Kasper Schmeichel but Rodallega's departure to injury did not end Fulham's hopes as Karagounis brilliantly drilled a 20-yard free-kick into the top corner with five minutes to go.
However, Dyer scored even later as he slid in to convert David Nugent's right-wing cross.