The Clarets have traditionally not done brilliantly in cup competitions in recent years but crucial goals from Jay Rodriguez and Kevin Long means they right in the mix in the last 16.
The visitors did look the brighter of the two sides early on in freezing conditions in West London with Matej Vydra and Phil Bardsley both testing Marek Rodak.
However, Burnley didn’t have to wait too much longer to edge out a breakthrough as Rodriguez headed home Jack Cork’s cross.
But despite finding themselves trailing after the restart, Fulham were no mood to roll over in this one as Bailey Peacock-Farrell produced a fine double stop Aleksandar Mitrovic.
And that little blip handed Burnley the perfect chance to go on and clinch their Fifth Round spot.
Rodriguez confidently made it 2-0 from the penalty spot with 20 to go before his clever knock down was
fired home by Long.
Full Time from Craven Cottage it’s Fulham 0 Burnley 3