Divock Origi and Mamadou Sakho scored in quick succession at the end of the first half to put Liverpool in control, and Everton's task got even harder in the second half when they had Ramiro Funes Mori sent off.
Daniel Sturridge and Philippe Coutinho added further goals for the rampant hosts, who enjoyed complete dominance in the second half and picked up a comprehensive victory that piles the pressure on Everton boss Roberto Martinez.
The match began at a frenetic pace and the hosts almost took the lead after just six minutes when Adam Lallana was played clean through on goal, only to be denied by a big save from Joel Robles when he only had the keeper to beat.
It was end-to-end stuff at times in the opening exchanges at Anfield, and Everton's threat on the break was highlighted when Kevin Mirallas beat three Liverpool players on his way forward before curling an effort over the crossbar having seen his initial pass towards Romelu Lukaku cut out.
There was a better chance for the Toffees seven minutes later when Mirallas picked up a pass from Gareth Barry before working space to shoot, but he went for power from the edge of the box and fired his effort well off target.
Coutinho then fired one over the top from 25 yards before Lallana again squandered a big chance when he steered his header narrowly past the post from Nathaniel Clyne's cross.
Liverpool began to gain an element of control over the contest as the first half wore on, and Everton had Joel to thank for keeping them in the game when he once again came out on top in a one-on-one situation, this time denying Roberto Firmino.
Origi, once again preferred to Sturridge, drew a more routine stop from the keeper towards the end of the half, but the Belgian did break the deadlock with only two minutes remaining until the interval when he climbed above John Stones to steer James Milner's cross in.
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One quickly became two as Liverpool took the game away from Everton in the closing stages of the first half, with Sakho being left unmarked to nod in another Milner cross from close range and double his side's advantage.
Things went from bad to worse for the visitors just five minutes after the restart when they were reduced to 10 men, with Funes Mori being shown a straight red card for a high challenge on Origi, who was forced off on a stretcher as a result.
It was the 21st red card the fixture has seen in the Premier League era - more than any other - but the first in the last nine meetings between the two sides.
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From that point on it was one-way traffic for Liverpool, who were queueing up to have a go at the Everton goal as the match threatened to turn into a rout.
Sturridge, on for the injured Origi, drew a routine stop from Joel before adding the third goal, collecting a pass from Lucas Leiva and applying a smart two-touch finish from inside the area.
The final half-hour of the game saw Liverpool camped inside the Everton half and creating chances almost at will, with Coutinho seeing one of the clearer openings blocked by Bryan Oviedo after Milner had again pounced on a loose ball out from the back.
The shot count continued to rise for the home side as Alberto Moreno, Joe Allen and Clyne all had efforts in quick succession before the fourth did arrive when Coutinho drilled a trademark finish into the bottom corner.
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The Brazilian threatened again moments later with a similar strike that this time Joel managed to save, while Moreno came close with a couple of powerful efforts as he looked to get in on the act.
Everton's misery was prolonged by seven added minutes, the first of which saw the visiting keeper called into action yet again as he kept out Jordon Ibe's deflected shot from range.
Clyne, Sturridge and even Lucas had further attempts in stoppage time, but Liverpool were unable to add to their lead despite having more than 40 efforts on goal throughout.
The victory sees Liverpool make it four wins in a row in all competitions and 11 consecutive Merseyside derbies without defeat in the league, moving them up to seventh in the standings.
Everton, meanwhile, are now without a win in seven games and must pick themselves up ahead of Saturday's FA Cup semi-final against Manchester United at Wembley.