It was Liverpool who started brightest and played with plenty of energy and tempo, and their pressure would soon pay off when Sadio Mane used his pace and power to run at the heart of Everton's defence and then slide a left-footed shot into the bottom corner in the eighth minute.
It was the home side that were in full control and were looking threatening and so it came as a surprise in the 28th minute that the Toffees found an equaliser against the run of play. Ashley Williams' corner broke in the box to Matthew Pennington who poked home the equaliser from close range.
Yet, it would soon go sour for the youngster when he allowed Philippe Coutinho to run at him three minutes later, turn inside on to his right foot from the left and then steer a superb effort into the net that allowed Joel Robles no chance.
After the break, Coutinho had a chance to make it 3-1 when he was nearly on hand to tuck home after Emre Can's strike was saved by Robles, though Williams was just about able to clear. A black mark for the Reds is that Mane had to be taken off after a collision with Leighton Baines, and was replaced by Divock Origi in the 57th minute.
Nonetheless, it would soon be 3-1 when the brilliant Coutinho again ran at Everton down the left side, and then fed in substitute Origi who fired home from the edge of the penalty area with Robles wrong-footed in the 60th minute.
Perhaps the visitors' best moment to get a goal back was through Mason Holgate in the 69th minute, though he saw his effort well saved by Simon Mignolet. There were few major alarms in the closing stages as Liverpool cruised to victory at Anfield.
The win lifted Jurgen Klopp's side seven points clear of fifth-placed Manchester United, who have three games in hand, including at home against West Brom later today.
Everton are seven points shy of the top four having played more matches than their rivals.