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The Clarets were handed an excellent opportunity to take the lead when Lukas Jutkiewicz pounced on Lee Cattermole's poor header. The striker teed up Marvin Sordell in the box, but he sliced his shot wide of the target from close range.
Sean Dyche's men continued to have the better of the early stages and David Jones went close as he drilled an effort from range with his left-foot, but Vito Mannone made the stop.
Sunderland went close when Adam Johnson managed to lift a cross into the box for the run of Santiago Vergini, and the defender's glancing header went just wide of the post.
In the second half, the Black Cats began to see more of the ball and Jack Rodwell threatened Tom Heaton's goal when he drove forward and powered a shot at the net, but the keeper parried the ball wide.
Sunderland almost cut the Clarets open at the back when Emanuele Giaccherini raided down the flank and he tried to cut the ball back to Rodwell, but Jason Shackell intercepted to clear the danger.
Scott Arfield tried to sneak his side into the lead when he tried to catch Mannone out at his near post, but the Italian was alert to the danger and managed to save the effort down to his right.
Jutkiewicz came close to firing his side ahead when he beat the challenge of John O'Shea and fired a low shot at goal from a tight angle but Mannone beat his effort away.
Ashley Barnes came within inches of handing his side a winner when his shot struck Cattermole and looped over Mannone only for the ball to hit the bar and bounce over.
Sunderland almost snatched the three points in the dying minutes when Patrick van Aanholt let fly from distance but his effort crashed off the post, hit Heaton in the legs and bounced agonisingly wide.