They took the lead in the first minute, Brian Deane and David Healy setting up Jermaine Wright to score with a low drive past goalkeeper Brian Jensen.
Burnley were gifted an equaliser after 10 minutes when Neil Sullivan's clearance was charged down by Lee Roche after Clarke Carlisle's error.
Burnley boss Steve Cotterill
"Their early goal steeled us. We had a determination to do well and we have to make sure we have that in every game.
"It's a great night for us and it was terrific from our lads.
"I thought we played some great football and arguably could have scored two more."
Leeds: Sullivan, Kelly, Carlisle (Lennon 67), Butler,
Kilgallon, Joachim, Wright, Gregan, Pugh (Simon Johnson 76),
Healy, Deane (Ricketts 57).
Subs Not Used: Carson, Walton.
Booked: Pugh, Butler.
Goals: Wright 1.
Burnley: Jensen, Duff, McGreal (Sanokho 80), Sinclair, Camara,
Grant, Roche, Hyde, O'Connor, Duffy (O'Neill 86), Blake.
Subs Not Used: Pilkington, Scott, Yates.
Booked: Sinclair, Duff.
Goals: Roche 10, Duffy 31.
Ref: P Dowd (Staffordshire).