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They took the lead with two goals in 30 seconds on the stroke of half time.
Kevin Campbell continued his good form with a headed effort from a Mark Pembridge free-kick.
The Welsh midfielder struck a sweet curling cross from the right of the pitch and Campbell rose above the defence to knock the ball into the net with the back of his head.
Then as the Goodison crowd was still celebrating Thomas Gravesen strode down the middle of the park and drilled home a great strike from 30 yards.
Edwin Van Der Sar couldn't get close as the ball flew in at rocket pace.
Fulham had started the game on top, Junichi Inamoto had an effort saved within two minutes and a minute later threaded through a fineball for Sylvain Legwinski.
Richard Wright saved but the clearance fell to Barry Hayles whose fine shot was cleared off the line by David Weir.
But as the game progressed the Londoners faded badly and Everton's aggressive approach frustrated the visitor's neat passing game.
Joseph Yobo comfortably dealt with their two strikers and Hayles was reduced to taking pot shots at Weir rather than the goal.
In the second half the Blues continued to dominate and could have had four if it wasn't for the post, Canadian striker Tomasz Radzinski was denied by the upright with two clear chances.
And as the game wound down Fulham's frustration came out and the game descended into a ten-man brawl that saw Campbell and substitute Andy Melville booked.