Fulham are 18th in the table on 19 points, six behind Newcastle United who occupy the final place above the relegation zone. Burnley moved up to 15th place, eight points above Fulham.
Scott Parker's team had grabbed the lead in fortunate fashion when Ademola Lookman's corner struck the chest of Ola Aina and Burnley substitute Robbie Brady missed the ball on the line.
Within three minutes, however, Burnley had drawn level when Jay Rodriguez broke down the right flank and slipped a perfect pass to Barnes who, despite a poor first touch, converted the chance.
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