Felix Magath's side, with just one win since New Year's Day, showed signs that have the stomach for a relegation scrap, but are five points adrift of safety after an own goal from David Stockdale and efforts from Kevin Mirallas and Steven Naismith sandwiched an equaliser from Ashkan Dejagah.
After a goalless first half, Everton were forced to withdraw the in-form Ross Barkley at the interval, but his replacement helped to break the deadlock six minutes after the re-start as a corner dropped to Naismith on the edge of the box and his hopeful drive through a crowd of players flicked off William Kvist, onto the unfortunate Stockdale and into the back of the net.
Fulham's response was impressive, with Dejagah stepping off the bench to lash home a leveller on 71 minutes as he cut inside off the left flank and arrowed into the top corner.
Everton's substitutes were to be the game-changers, though, and Mirallas restored their lead after latching onto a threaded pass from Aiden McGeady and Naismith wrapped up the points three minutes from time as the ball broke his way off Leighton Baines and he steered into the bottom corner from close range.
Magath took the gamble of starting 17-year-old French striker Moussa Dembele and the teenager showed enough promise to suggest he was worthy of the German's faith.
Dembele was little more than a spectator as Fulham made a lively start, Everton clearing a dangerous ball from John Arne Riise before Brede Hangeland sent a looping header narrowly wide.
A glimpse of the ability that convinced Magath to start Dembele was evident in the 22nd minute when the French teenager raced clear and supplied a perfect pass to Kieran Richardson.
The vision he showed deserved a better outcome than Richardson's weak shot that meant the best move of the half went unrewarded.
A rare moment of panic in Fulham's defence came when Deulofeu crossed but the ball sped to safety with no Everton player able to reach it in time.
The final chance of the first half arrived just seconds before the interval, but Dembele was unable to convert Lewis Holtby's delicate chip.
Fulham were left to rue the missed opportunity as just six minutes into the second half Everton edged ahead with a large slice of luck forcing the opener.
A corner reached Naismith and he drilled a low shot towards goal from the edge of the penalty area. The ball struck Fulham midfielder Kvist before being diverted into the net by the blameless Stockdale, who had been deceived by its sudden change in direction.
Everton went within a whisker of extending the lead when Romelu Lukaku narrowly missed Leighton Baines' cross.
Just as Fulham looked to be in danger, they responded with Dejagah's superbly taken goal that saw the Iranian cut inside two defenders from the left flank and beat Tim Howard at the near post with a powerful 20-yard effort.
Inspired by Dejagah's brilliant intervention, the Cottagers pressed for a second goal with Johnny Heitinga and Pajtim Kasami denied by successive saves from Howard.
Instead, the next goal came from Everton with Howard pumping the ball upfield to McGeady, who released Mirallas with a precise pass.
There was still plenty of work for the Belgian forward to do, but he held off Hangeland and planted the ball into the bottom left corner.
Fulham's defence was in disarray and could have conceded a penalty when Heitinga hacked down Mirallas, but the ball still squirted to Naismith who put the result beyond doubt.