Lydon Dykes missed a golden opporunity in normal time before VBobby Decordova-Reid and Neeskens Kebano fired the visitors into the next round.
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.
QPR started strongest but Fulham 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.
The striker was played in by Josh Onomah and bore down on goal, only for Yoann Barbet to make a brilliant last-ditch block. On the hour mark, Mitrovic was denied on the goal line after good work by Ivan Cavaleiro.
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.
Additional reporting by Press Association.