Jose Mourinho restored Shaun Wright-Phillips to the starting line-up, gave an outing to Glen Johnson at right-back and let loose Carlo Cudicini between the sticks, but still the Premiership champions showed no signs of taking their foot off the pedal.
Goals from Arjen Robben, Frank Lampard, Herman Crespo and from captain John Terry sealed an empathic victory to ensure a home tie against Colchester United in the fifth round.
]The goal-fest started after concerted pressure by the Londoners on 22 minutes and it was an excellent counter-attacking play started by a swift change of play by Lampard.
He fed Eidur Gudjohnsen on the left. The Icelandic player lofted a through ball into Crespo, who showed deft skill to chest the ball down and tap it into the path of the on-rushing Robben.
The Dutch winger took one touch to control the ball before sliding an inch-perfect ball past the hapless reserve goalkeeper Iain Turner.
Turner was a late call-up after first choice Richard Wright turned an ankle during his pre-match warm up.
Thirteen minutes later and it was all but over as a contest.
Wright-Phillips wriggled into the Everton penalty area past two defenders before being upended by Nuno Valente. Up stepped Lampard to dispatch the resultant penalty with consummate ease, shooting low and wide of Turner's dive.
By half-time, Everton must have being wishing the game was over, because they conceded a third goal five minutes before the break. This time it was training pitch type of goal.
Lampard zipped in a free-kick from the left into the six-yard area, Crespo was first to react and glanced the ball past Turner.
Everton, in the second half decided to make a game of it and managed to get a goal back.
It took them until 60 minutes to register their first effort on goal, a Tim Cahill free-kick, straight into the midriff of Cudicini.
But their consolation goal came on 72 minutes via the penalty spot.
Robert Huth handled a fierce Cahill drive and Mikel Arteta slotted home easily.
John Terry lashed home a marvellous right-footed piledriver from eight yards out following a clearance from a Huth free-kick to ensure the Londoners progressed into the next round.