All White Chris Wood scored one of Burnley's goals.
The loss left Fulham in 18th spot on 27 points, 10 behind Southampton with three games left to play, while Burnley rose from 17th to 14th on 39 points.
Despite plenty of possession, Fulham failed to create many decent scoring chances and return directly to the Championship after a single season in the top flight, where they will join West Bromwich Albion, who were relegated at the weekend after losing to Arsenal, and bottom side Sheffield United.