The Premiership side enjoyed the bulk of the fifth round contest but rarely troubled Clarets goalkeeper Marlon Beresford.
Burnley enjoyed the perfect start as they sought to become the latest team to pull off a giantkilling act in this season's competition.
Hesitancy in the Fulham defence enabled Alan Moore to race through the heart of the Cottagers' rearguard before slotting home after just four minutes.
Louis Saha wasted an excellent chance to draw level but the in-form Steed Malbranque salvaged a replay for the hosts with a precise shot on the stroke of half-time.
In the second period, Fulham continued to ask the most searching questions but could not find a route through the stubborn visiting defence and a trip to Turf Moor is now on the agenda for Jean Tigana's side with a quarter final berth at stake.
