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The win moves David Moyes' side seven points clear of seventh-place Wigan, who play on Sunday, and they remain three points behind Arsenal, who are fifth.
Everton have kept clean sheets in their last six games in all competitions, taking 13 points from the last 15 available.
Brown's men worked hard but were outclassed by an Everton side who started brightly and probed tirelessly for a breakthrough.
Despite the absence of a recognised striker, lone front man Tim Cahill was ably supported by Leon Osman, Steven Pienaar and Fellaini as the hosts dominated possession.
Hull's 4-5-1 formation indicated a determined rearguard action would be the order of the day and for 18 minutes they held firm impressively.
And in Marlon King, Bernard Mendy and former Everton forward Nicky Barmby - heckled throughout on his first return to Goodison Park having left the club for Liverpool in 2000 - they looked to catch the Toffees on the counter-attack.
Everton cemented their place in the top six as goals from Marouane Fellaini and Mikel Arteta condemned Hull City to their fourth successive league defeat.
The home side could have doubled their lead in the 24th minute when Pienaar broke free on the right and squared dangerously across the face of goal, but none of his team-mates was on hand to convert.
Arteta troubled Hull throughout and he made it two with a spectacular free-kick on the stroke of half-time.
Hull never threatened a fightback as Everton easily held their lead.
Cahill then crossed for Fellaini to head narrowly wide and at the other end Michael Turner had Hull's only half-chance of the match, a tame shot dragged wide.
Hull, on the other hand, have won just one of their last 12 games and are without a point since drawing with Liverpool on 13 December.Manager Phil Brown will be furious that Fellaini's goal was allowed to stand but could have no complaints at the result given Everton's superiority in a physical encounter.