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Liverpool were never under any serious or sustained pressure and scored twice late in the first half through Divock Origi and Mamadou Sakho, both from James Milner crosses, before Daniel Sturridge came off the bench to make it 3-0 and Philippe Coutinho finished off the rout.
Sturridge was introduced in place of Origi, who was taken off on a stretcher after Funes Mori planted his studs into the Belgian striker's ankle, which earned the Argentine a straight red card despite his protestations of innocence to the referee.
Liverpool, who are now unbeaten in the last 11 top-flight Merseyside derbies, stay seventh in the table, while Everton, who face Manchester United in the FA Cup semi-final on Saturday, are 11th.