Captain Steven Gerrard set the home side on their way with the opener and two quick-fire Daniel Sturridge goals put Liverpool 3-0 to the good by half-time.
Luis Suarez got in the act with Liverpool's fourth goal early in the second half and things could have been even worse for Everton as Sturridge fluffed the chance to net a hat-trick when he missed from the penalty spot, but it did not matter in the end as Brendan Rodgers recorded his first victory in a Merseyside derby to see his side go four points clear of their local rivals in the table.
Everton came flying out of the blocks and the returning Ross Barkley went close to scoring after just two minutes when he picked up Kevin Mirallas' pass and raced to the edge of the box before drilling his left-footed shot just over the crossbar.
Liverpool fashioned their first chance of the game a minute later when the ball dropped invitingly for Jordan Henderson 25 yards from goal and his volley was beaten away Tim Howard.
Suarez forced Howard into a good save with a snap-shot from the edge of the box which the USA international did well to get down to turn round the post.
Howard denied Liverpool the opener on 18 minutes when Sturridge slipped Raheem Sterling in on goal, but the Everton keeper stood up well to keep out the winger's effort with his feet.
Liverpool made the breakthrough their early dominance deserved on 21 minutes when Reds skipper Gerrard rose highest to power home Suarez's corner with Barkley on the post unable to keep it out.
In a double blow for Everton they lost Romelu Lukaku to injury after he collided with team-mate Barry from the corner and was replaced by Steven Naismith.
Sturridge went close on 27 minutes when Suarez teed him on the edge of the box and the striker curled his right-footed effort just over the top.
Everton forced Simon Mignolet into his first serious save on the half hour when the ball fell to Phil Jagielka inside the box and his sliding shot was well kept out by the Liverpool keeper with Barry blazing his follow up effort wide of the target.
The dangerous Kevin Mirallas was inches away from an equaliser on 32 minutes when he went past Gerrard and fired his shot just wide of the far post.
However, 60 seconds later Liverpool doubled their lead as Philippe Coutinho slid Sturridge in on goal and the England man kept his cool to slot it past Howard.
Things got even better for Liverpool just two minutes later when Sturridge broke the Everton offside trap to latch onto Kolo Toure's pass over the top and the striker produced a sublime lofted finish over the onrushing Howard to send the home fans into raptures.
Everton came out for the second half in determined fashion and substitute Leon Osman went close to scoring when he found space inside the box and curled a shot towards the top corner which was superbly kept out by the flying Mignolet.
But almost immediately play switched straight down the other end and Suarez capitalised on a mistake by Jagielka to race clear from the halfway line and he kept his composure to slide his left-footed shot past Howard for his 23rd goal of the season.
Liverpool won a penalty on 54 minutes when Sterling latched onto a ball into the box and the winger was brought down by the onrushing Howard.
Sturridge stepped up to complete his treble from the spot, but the striker blazed his penalty horribly over the crossbar to let Everton off.
Coutinho went close on the hour mark when his shot flicked off the outside of the post as Liverpool laid siege to the Everton goal.
Everton tried to grab a consolation and James McCarthy was denied by Mignolet as Liverpool ran out convincing winners to boost their top four hopes and record their biggest derby win since 1982.