The Premier League outfit needed extra time to see off their west London neighbours 2-0, after Lyndon Dykes had missed a gilt-edged chance to win it for Championship Rangers.
Both these sides, struggling in their respective divisions, were looking to shake off the ring rust following an unplanned winter break.
Rangers' last Championship fixture fell foul of Covid-19 while an outbreak at Fulham meant Scott Parker's side had not played since Boxing Day.
Parker nevertheless named a strong side, but it was Rangers who should have taken an early lead when Macauley Bonne headed wastefully over from a corner.
But the Cottagers gradually took control and almost took the lead after a bit of quick thinking from Joe Bryan following a hashed clearance by Seny Dieng.
Bryan's first-time lob from 30 yards looked to be creeping in but to the Rangers keeper's relief the ball dropped narrowly wide.
Aleksander Mitrovic, back in the Fulham starting line-up following his breach of coronavirus restrictions on New Year's Day, had a chance to make amends as halt-time approached.
Rangers remained a threat, though, and Bright Osayi-Samuel's fierce, swirling drive brought another fine stop from Rodak.
Ivan Cavaleiro should have given Fulham the lead on the hour when he was sent through by Onomah but for some reason the winger checked his run on goal and tried to cut inside the chasing Rangers defence. Cavaleiro eventually squared the ball to Mitrovic, whose low drive was hacked off the goal-line by Barbet.
At the other end Rodak got a touch on Dominic Ball's cross-shot with Dykes unable to apply a finish.
The same player had a wonderful opportunity to win it in normal time when a simple ball over the top sent him through – but he rushed his shot and blazed over.
In extra time Kebano stood up a cross at the far post from where Decordova-Reid headed wide.
But Decordova-Reid came up with a 103rd-minute opener after Antonee Robinson crossed from the left, the forward playing a one-two with Mitrovic before rifling home.
Moments later the ball fell to Kebano in the area to lash home and ensure Fulham's passage into round four.