Scott Parker's side would have climbed out of the relegation zone for the first time since 20 December if it had managed a rare home victory.
But Patrick Bamford put Leeds in front in the first half and, although Joachim Andersen equalised before the break, Raphinha punished sloppy Fulham defending to win it in the second half.
Third-bottom Fulham remains two points behind fourth-bottom Newcastle United, which has two games in hand including a crucial clash with fellow struggler Brighton and Hove Albion on Sunday (AEDT).
Beaten in three of its past four games, Fulham has now lost 10 home league matches this season.
Survival is still possible given Newcastle's struggles, but with only eight matches left Fulham is running out of time to avoid an immediate return to the Championship.
Mid-table Leeds's first win in four games was its first in London in 17 attempts.
It appeared to have made the perfect start in the eighth minute when Luke Ayling looped his header into the far corner from Tyler Roberts's delivery.
But VAR (video assistant referee) disallowed the goal after seeing that Roberts's arm was narrowly offside before he crossed into the Fulham area.
Fulham's Josh Maja forced a superb save from Illan Meslier with a fierce close-range effort.
Leeds made the most of that escape to take the lead in the 29th minute.
A quick throw caught Fulham sleeping and Jack Harrison found space for a cross to the near post, where Bamford made a perfectly timed run to smash past Alphonse Areola from close range.
It was Bamford's 14th goal of the season but he nearly gifted Fulham an equaliser with a miscued headed clearance that flew just over moments later.
Despite just two goals in its previous nine home league games, Fulham responded well and equalised by exposing Leeds's long-standing set-piece weakness in the 38th minute.
When Marcelo Biela's side failed to deal with Ademola Lookman's corner, Danish defender Andersen got in front of Ayling to volley home from six yards.
Andersen's first goal since his arrival on loan from Lyon last year was the 15th conceded by Leeds from a corner this term.
Meslier kept out Andre-Frank Zambo Anguissa's stinging blast just before half-time.
Having showcased his predatory instincts earlier, Bamford turned provider in the 58th minute.
Kalvin Phillips robbed Mario Lemina and Bamford slipped a deft pass through to Raphinha, who punished poor Fulham defending to poke his shot past Areola.
Unable to muster another equaliser, Fulham was left to rue VAR's refusal to award a penalty after Bamford's arm made contact with the ball while he jumped to clear a free-kick.