Josh Maja’s brace – his first goals for Fulham – gave them a superb away win and raised hopes they might yet escape relegation.
Recent signing Maja, making his full Premier League debut, opened the scoring four minutes into the second half.
Ola Aina burst down the left and exchanged passes with Ademola Lookman before laying the ball across for striker Maja to score from close range.
And Maja was on hand for another tap-in on 65 minutes, following up after Harrison Reed’s shot had hit the post.
The result – Fulham’s first-ever league win at Everton – cut the gap between the third-from-bottom Whites and Newcastle, the team above them, to seven points. The Magpies play Chelsea on Monday evening.
Fulham produced an excellent performance at Goodison Park, where they were by far the better team and almost took an early lead when Lookman’s left-wing corner was flicked against the far post by Bobby Decordova-Reid.
The rebound dropped to Aina, who was unable to force the ball home.
Lookman, playing against his former club, and Reed both fired narrowly wide as Fulham continued to dominate.
Everton themselves hit the woodwork when Seamus Coleman strode forward just after the half-hour mark and saw a left-footed effort bounce off the post.
That was a rare attack from the home side, who were on the back foot for most of the game.
Fulham have dropped crucial points this season after failing to find the net despite some excellent play.
It seemed like an all-too familiar story was unfolding when they went in at the interval goalless after controlling most of the first half.
But in Maja, their loan signing from Bordeaux, they might have found the poacher they have so desperately needed.
He worked tirelessly up front and his predatory instincts delivered three priceless points.
Fulham: Areola, Tete, Andersen, Adarabiayo, Aina, Lemina (Onomah 90), Reed, Loftus-Cheek, Reid, Maja (Cavaleiro 72), Lookman (Anguissa).
Subs not used: Rodak, Hector, Odoi, Ream, Kongolo, Robinson,