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Ashley Barnes' second-half goal stopped Fulham from moving within four points of safety as Burnley earned a 1-1 draw at Turf Moor.
There were few clear-cut chances in the first half until James Tarkowski's left-wing cross narrowly evaded Jay Rodriguez after a touch by Fulham goalkeeper Alphonse Areola.
Fulham went ahead on 49 minutes when Ola Aina chested in after Joachim Andersen's flick-on from Ademola Lookman's corner.
Burnley got back on level terms just three minutes later when Rodriguez picked out Barnes, who fired beyond Areola.
Josh Maja and Ruben Loftus-Cheek came closest to earning a second successive win for Fulham, who remain in 18th, six points adrift of 17th-placed Newcastle United.
Burnley move up to 15th on 27 points, eight above Fulham.
See: Burnley report | Fulham report
Next fixtures
Burnley: 20 Feb v West Brom (H)
Fulham: 20 Feb v Sheff Utd (H)
Match officials
Referee: Jonathan Moss. Assistants: Marc Perry, Nick Hopton. Fourth official: Martin Atkinson. VAR: Darren England. Assistant VAR: Dan Robathan.
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