© PA Photos
The home side had the majority of the possession early in the game, but a series of crosses into the box failed to connect with Sam Vokes, who was the only forward named in the starting XI.
The visitors broke the deadlock in the 27th minute, with Kim Bo-Kyung picking the ball up in midfield and feeding it through to Craig Conway on the edge of the penalty area, allowing the Scot to blast the ball past Lee Grant.
The league leaders had several chances to double their lead before half time, but Burnley got numbers behind the ball to ensure that the deficit was just one goal at the break.
The hosts were lucky not to be reduced to 10 men, as Ross Wallace appeared to hit Kim in the face with his arm, leaving the South Korean needing treatment to his nose.
The visitors looked to extend their lead straight after the restart, as Conway had an effort blocked inside the opening two minutes.
© PA Photos
Sean Dyche's side began to find their way into the match, with Kieran Trippier creating several chances, before Vokes forced a save from David Marshall and Martin Paterson headed the ball high of the crossbar.
Marshall was needed to make another stop inside the final five minutes, as Danny Ings tried to force an equaliser for the hosts, but the Scottish keeper could not keep David Edgar's stoppage-time header out of the net.
The result leaves Burnley four points clear of the relegation zone.