The home side started the game well and came within inches of taking the lead in the fifth minute, when George Boyd fired a volley from the edge of the area that cannoned off the crossbar and went over.
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Burnley thought they had taken the lead as Danny Ings released the run of Lukas Jutkiewicz through the middle and the striker slotted the ball past Adrian, only to see his effort chalked off for offside.
The Clarets were dominating the game and threatened again when Boyd and Kieran Trippier combined down the right to tee up Jutkiewicz in the middle, but the striker guided his header just over the bar.
The visitors continued to press and went even closer when Stewart Downing cut in from the right, but he saw his left-footed effort clatter off the post and bounce clear.