The Premier League outfit broke the deadlock at the Lamex Stadium in the fifth minute when Steven Naismith poked home from close range after home goalkeeper Chris Day had failed to collect the ball.
However, the Toffees were dealt a blow midway through the first half when Bryan Oviedo was substituted with a leg injury and was taken to hospital.
That did not deter the Merseysiders from making it 2-0 just past the half-hour mark in the fourth-round tie, though. Naismith was again on target as he pounced on a mistake from Jon Ashton to run through on goal and slot the ball home with ease.
Johnny Heitinga added the third for Everton nine minutes into the second half when he headed home from close range.