Going into the game with high expectations, this was a frustrating night for Martin Jol’s team who failed to build on last Saturday’s stunning defeat of Newcastle United.
As expected it was the hosts that enjoyed the majority of the early possession, with Brede Hangeland doing well to snuff out a dangerous run from Denis Stracqualursi and Hughes clearing one Leighton Baines cross to safety.
But Fulham looked to have settled well, with Johnson linking well with both Bryan Ruiz and Damien Duff as the Whites threatened the hosts on the counter.
And it was from one such break forward that saw Fulham take the lead, with Johnson teasing a ball into the Everton box, from which Chris Baird saw his strike at goal handled by John Heitinga.
Danny Murphy stepped up and did what few do better for his second goal from the spot in seven days - much to the delight of the travelling Fulham faithful.
The home side hit back, Baird doing well to block a chance for Magaye Gueye, while Tim Cahill moved David Stockdale into an impressive stop.
Marouane Fellaini went close too, while Shane Duffy thought he had scored, only to see Hughes make an incredible headed clearance.
However, Everton had the momentum, and it wasn’t long before David Moyes’ side levelled as Landon Donovan crossed from the right and Stracqualursi rose high to head confidently beyond the reach of Stockdale.
A powerful drive from Donovan stung the hands of the Fulham ‘keeper late in the half, and Gueye could well have sent Everton into the break in front, but much to the delight of the Whites’ back-line the Frenchman shot wide.
Both sides made a bright start to the second-half, and the Whites created the first decent chance of the period when Ruiz delivered an inviting cross on to the head of Dempsey – unfortunately last weekend’s hat-trick hero couldn’t find the target.
For the best part of the half, neither side really imposed themselves on proceedings, with one shot from Dempsey rising high and Johnson firing one low drive at Howard.
As the game ebbed and flowed, Everton were certainly there for the taking, but when Cahill and Donovan went close, the hosts stepped back into gear.
And it was from the American’s free-kick from the right that the Toffees took the lead, Fellaini meeting his pin-point delivery with a powerful header.
Cahill went close to a third moments later, but Stockdale showed incredible reaction to tip the ball to safety.
At the other end, Fulham looked frustrated in their search for an equaliser, although the introduction of Academy graduate Marcello Trotta alongside Bobby Zamora suggested they would go for it.
As the game came to an end Ruiz went close from the edge of the Everton box to offer a moment of hope, but the hosts looked more than determined in their pursuit of passage into the Fifth Round.