Martin Jol took a step towards alleviating the pressure he has come under of late by overseeing an impressive turnaround courtesy of two expertly-taken goals
Steven Naismith gave the visitors the lead in the first half as they dominated the early proceedings at Craven Cottage.
But Fulham progressed to the next stage of the competition thanks to goals after the break from Dimitar Berbatov and Darren Bent.
|VIEW FROM CRAVEN COTTAGE
|By Liam Twomey
Boy, did Fulham need this. A first home win in seven matches could not have looked less likely after a listless first-half performance devoid of confidence or imagination, but second-half strikes from Dimitar Berbatov and Darren Bent have provided a much needed respite for beleaguered boss Martin Jol, as well as a first win over Everton in nine attempts.
Roberto Martinez's men started in confident mood, and Steven Naismith gave them a deserved lead. But the visitors then inexplicably dropped deep, handing the initiative to Fulham and inviting the comeback.
That Gerard Deulofeu and Romelu Lukaku were the most dangerous attackers on the pitch will only add to Everton manager's frustration, but he must now concentrate on maintaining his side's unbeaten record in the Premier League.
Jol made six changes to the side that lost 2-0 at Chelsea on Saturday, with Berbatov returning from a hamstring injury.
Adel Taarabt and Giorgos Karagounis were among the other inclusions, as Aston Villa loanee Bent dropped to the bench, while Everton also made significant alterations to their side, with manager Roberto Martinez making eight changes.
Only Naismith, Coleman and Sylvain Distin retained their place for Everton, as Martinez handed a first start to loan signing Romelu Lukaku.
Everton made the brighter start, with Stockdale needing to save efforts from Naismith and John Heitinga inside the opening eight minutes.
Naismith, after testing Stockdale again a moment earlier, gave the visitors the lead in the 12th minute when he ran on to Gerard Deulofeu's pass and fired through the legs of the goalkeeper.
Fulham's first chance of note fell to Berbatov in the 17th minute but the Bulgarian forward could only head wide from Elsad Zverotic's cross from the right.
At the other end, Distin had a header cleared off the line by Zverotic before Heitinga volleyed over from close range, while Berbatov failed to direct another header on target.
Naismith almost had a second goal on the half-hour mark, but the Scotland international fired into the side netting after combining with Deulofeu again.
The Barcelona loanee was proving a constant menace to the Fulham defence, driving forward before feeding Lukaku, but he saw his effort tipped away by Stockdale.
And Deulofeu was in the action again as the half drew to a close, twice drilling low shots that had to be dealt with by the busy Stockdale.
Fulham began the second half with greater urgency and Alexander Kacaniklic tested Everton keeper Joel Robles with a firmly hit drive just after the restart.
Berbatov levelled matters in the 54th minute, tucking home from 10 yards after Coleman's challenge on Taarabt directed the ball into his path.
And Bent fired them in front just four minutes after coming off the bench, collecting Karagounis' disguised free-kick and firing past Robles in the 68th minute.
Coleman fizzed an effort just wide five minutes later as Everton searched in vain for an equaliser, while Deulofeu and Lukaku also went close.
And Coleman had another chance to draw Everton level, only for Stockdale to deny him, as Fulham claimed victory.