The Toffees raced into an early two-goal lead against their League Two opponents before adding a third in the second half to reach the fifth round for the fifth time in the last six seasons.
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It took less than two minutes for Everton to get their first of the afternoon, and it was far too simple for the visitors as Seamus Coleman's pass over the top released Aaron Lennon, who rolled the ball across the six-yard box for Arouna Kone to tap home from close range.
The early goal dampened the atmosphere inside Brunton Park, but belief began to grow a little more when two Everton defensive errors handed the home side a couple of sights of goal inside the opening 10 minutes.
Both fell to Derek Asamoah, who scored against Everton's Merseyside rivals Liverpool at Anfield in the League Cup earlier this season, but the striker first fired wide after Tom Cleverley had sold Phil Jagielka short, and then saw a stinging half-volley beaten away by Joel Robles following a mistake from Ramiro Funes Mori.
The Toffees took a major step towards the fifth round with their second goal after just 14 minutes, however, as Bryan Oviedo exchanged passes with Ross Barkley before firing a low cross in to Lennon, who took a touch before slotting his finish underneath the keeper.
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A period of Everton domination followed before Carlisle began to gain a foothold in the match once again, but it wasn't until four minutes before half time that either side had another sight of goal, when Kone scuffed a volley comfortably into the arms of Mark Gillespie.
The rare openings that were being created by the hosts all fell to Asamoah, and he was released by Hallam Hope in the second half, only to see his eventual effort deflected high and wide of the target.
It took a couple of important blocks from Michael Raynes and David Atkinson either side of the hour mark to deny Coleman and Barkley respectively, but the third goal arrived shortly afterwards to put the game to bed once and for all.
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It was Barkley who got it, picking the ball up outside the area before working space to shoot and slamming a deflected effort past a statuesque Gillespie into the top corner.
Cleverley came close to a fourth with a curling effort towards the bottom corner, and just a minute later Kone's drilled ball from an impossible angle clipped the post and flew right across the face of goal.
The last laugh almost went to the home side, however, as they had two chances late on. The first fell to Antony Sweeney, who was denied by Joel from a tight angle inside the box, before Alex Gilliead's good work down the right set up an opening for Jabo Ibehre, whose effort was blocked by Jagielka right in front of the keeper.
There was to be no goal for the hosts, however, as they became the final League Two side to be knocked out of this season's competition, while Everton march on to the last 16.