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.
1. Bailey Peacock-Farrell 7
An important clean sheet for the Northern Irish stopper, who made a number of impressive saves. Denied Bryan at full stretch after the break, immediately denied Mitrovic twice and then made a big stop to deny Kamara.
2. Phil Bardsley 8
Solid display from the Burnley right back. Strong in the challenge, kept things tight, defended near post crosses effectively, and utilised the channels well to get the Clarets in behind.
3. Erik Pieters 6
Couldn't fault the Dutchman. Ran a tight ship down his side of the pitch, handled Cordova-Reid very well and was neat and tidy in possession.
4. James Tarkowski 7
Another commanding display at the back for the Clarets. Dealt with the pace and physicality of Kamara and the aerial threat of Mitrovic. Made a vital block to keep out Bryan's effort in the first half.