First half goals from Tim Cahill and Andy Johnson put the home side in a comfortable position, with Liverpool struggling to hit the target, despite numerous shots on goal.
Steven Gerrard hit the woodwork twice, two penalties were dismissed by referee Graham Poll and, to make matters worse for Rafael Benitez, John Arne Riise was stretchered off with an ankle injury.
The result leaves Liverpool on four points after three games, while Everton hit the heady heights of the top of the table, albeit until 4.45.
