Martin Jol took a step toward 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.
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 its side, with manager Roberto Martinez making eight changes.
Only Naismith, Coleman and Sylvain Distin retained their places 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 onto 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 defense, 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 leveled 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 equalizer, 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.