The visitors were quick out of the starting blocks but they conceded in the eighth minute against the run of play, Anthony Pilkington curling a free kick into the Fulham net via a Lucas Piazon deflection.
It did not take long for the Cottagers to equalise, however, with just six minutes elapsing before a passage of play involving Tom Cairney, Sone Aluko and Ryan Fredericks culminated in the ball being cut back to Stefan Johansen, who powerfully sidefooted home.
However, the away side continued to press on and were rewarded in the 33rd minute via Ryan Sessegnon, who bundled the ball home on the rebound from a yard out following Fredericks's impressive foray down the flank and Cairney's initial bar-bound effort.
The second half saw Fulham go down a gear as Cardiff continued to struggle in terms of troubling their opponents, and Slavisa Jokanovic's charges could even afford to miss a number of additional chances.
Bluebirds goalkeeper Brian Murphy was tested by Piazon and Aluko, but they did not prove crucial come the final whistle as Fulham cruised into the fourth round of the competition.