Leeds were doing all the early pressing and came inches from taking the lead when a poor backpass was picked up by Ross McCormack and he chipped Heaton, only to see his effort clip the post and bounce clear.
Brian McDermott's side made their dominance pay as Matt Smith flicked a throw-in into the path of Ross McCormack, who headed the ball past Heaton into the net to hand his side the lead.
However, Burnley got an equaliser just before the break to peg Leeds back as Dean Marney lifted a ball over the top of Kieran Trippier and his low cross into the box was turned into his own net by Jason Pearce.
After half time, Ross Wallace had the chance to fire his side into the lead but his volleyed effort went just wide of the post from outside the penalty area.
Leeds responded and Heaton had to do well to stop a powerful effort from Connor Wickham finding the net, as his shot from out wide was heading for the top corner before the keeper took action.
Burnley took the lead with a great individual effort from Arfield as he picked up the ball and, after seeing his first effort blocked, he fired his second into the top corner beating Jack Butland.
The visitors tried to press late on but the home side managed to hold out to secure the three points.