Ademola Lookman scored his second goal for the club after a brilliant counter-attack from the away side, before Ivan Cavaleiro broke Fulham's dreadful penalty run to make it 2-0.
Harvey Barnes halved the deficit with five minutes remaining, but it was too late for the Foxes to snatch a point.
The win lift's Scott Parker's side out of the relegation zone.
Story of the game
The contest started exactly as expected, with the home side enjoying large spells of possession and tiring Fulham.
Brendan Rodgers' side should have opened the scoring through both Youri Tielemans and Wesley Fofana.
The Belgium midfielder saw his looping volley bounce off the post before Fofana failed to keep his rebound down from six yards out.
Against the run of play, the Cottagers took the lead just before the half-hour mark.
Andre Zambo Anguissa broke through the line of Leicester's midfield before slotting Lookman through one-on-one with Kasper Schmeichel, before dispatching the ball past the Danish goalkeeper.
Things went from bad to worse for the Foxes when VAR penalised Christian Fuchs for a foul on Bobby Decordova Reid in the penalty area, providing Fulham with a chance to double their lead from the penalty spot.
Despite missing three penalties so far this season, including two in their previous two games, Cavaleiro converted from the spot to put the visitors two goals ahead.
Harvey Barnes and Cengiz Under were both introduced at the break, with Leicester switching to a 4-2-3-1 formation.
However, the Foxes' struggled to break the resolute Fulham defence in a second-half which lacked many clear cut chances for the Foxes.
They did grab a consolation through Barnes in the 85th minute after Jamie Vardy played the ball to the winger to fire past Alphonso Areola.
Takeaway from the game
Harvey Barnes must start for the Foxes
Despite struggling away at Liverpool last Sunday, Barnes impressed in the 3-3 draw with SC Braga on Thursday before being left out of the starting line-up at King Power Stadium.
In the first half, with Rodgers setting his side up with five at the back, it was evident that they were missing an offensive outlet out wide when attacking.
The system simply did not work, with Fulham simply playing one ball through Tielemans and Nampalys Mendy before being in a position to cause Leicester problems.
Barnes and Under were put on at half-time, but it will be something for Rodgers to work on in the future games as it's another home game against a side they should be beating where they have dropped points.
He showed why he should be in the starting line-up when he scored his third league goal of the campaign.