Jay Rodriguez scored twice to give the Clarets the advantage over Fulham before defender Kevin Long sealed the away side's passage.
The striker opened the scoring in the 31st minute when guiding his header from a Jack Cork cross beyond Marek Rodak.
He then slammed his penalty home after the break when Matej Vydra was brought down by Michael Hector inside the box.
And Long capped off a fine performance when firing the ball home from close range in the final 10 minutes of the tie.
Here are our RATINGS...
He then slammed his penalty home after the break when Matej Vydra was brought down by Michael Hector inside the box.
5. Kevin Long 7
A goal and a clean sheet for Burnley's "Mr Reliable". Made a number of key interceptions inside the penalty area, read the play well, gave away very little to Kamara and Mitrovic, strong in the air and took his goal well.
6. Johann Berg Gudmundsson 6
An invaluable 90 minutes in the bag for the Icelandic international. Slightly wasteful in front of goal with a number of key chances - a couple from open play and a set-piece - but kept the ball alive for Burnley's third.
7. Dwight McNeil 6
Sliced Fulham's defence open with several incisive passes, protected the ball well, attacked the space alongside Hector, but his deliveries were just off their intended target.
8. Jack Cork 8
A magnificent display in the middle of the park from Cork. Controlled the play, anticipated the home side's passes out from the back, broke up play nicely and battled well. Supplied a sumptuous cross from Rodriguez's opening goal.