James Pearce on an afternoon the Reds strengthened their top-four claims by putting noisy neighbours in their place
Jurgen Klopp approached the triumphant Kop to deliver a flurry of fist pumps.
The beaming Liverpool boss knew that this glorious victory was about so much more than just Merseyside pride.
His depleted Reds line-up had a taken a big step towards achieving their ambition of Champions League qualification. And they had done it in style.
Talk of an Everton revival and potentially leapfrogging the Reds was silenced. Liverpool's top-four push received the perfect shot in the arm while the Blues' lies in tatters.
Benjamin Franklin once said there were only two certainties in life: death and taxes. But Liverpool outshining their neighbours on derby day can't be far off being added to that list.
Everton's long wait for a first win at Anfield this millennium goes on. Kevin Campbell's place in pub trivia remains as secure as ever.
Yet again the high-stakes nature of this fixture brought out the best in Liverpool. Yet again the Blues folded under the pressure of the occasion.
But for the three first-half minutes that separated Philippe Coutinho restoring the Reds' lead after Matt Pennington had cancelled out Sadio Mane's opener, the outcome was never in doubt.
Divock Origi's emphatic finish settled proceedings as Klopp became the only Liverpool manager in history to win his first three derbies.
Forget Ronald Koeman's bizarre assessment about Everton controlling the game.
The reality is they were comprehensively outclassed as Liverpool cashed in on the key personnel the visitors were missing.
But for referee Anthony Taylor's generosity, the margin of victory would surely have been greater. It will remain a mystery how Ross Barkley remained on the field after an X-rated lunge on Dejan Lovren before the break.
Thankfully, the Liverpool centre-back avoided serious injury but victory did come at a cost with Mane limping off with a knee problem in the second half.
“I'm okay,” said the Senegal international as he walked gingerly through the mixed zone pulling a suitcase. A scan will determine the extent of the damage.
This wasn't a day for angry recriminations, it was a time for Kopites to bask in the glow of a performance brimming with desire and commitment in all departments.
'Mr Liverpool', who was remembered before kick-off with an emotional minute's applause and a mosaic which read 'Bugsy', would have nodded in approval at the sight of Klopp's men tearing into Everton with such a show of force.
The Reds made light of the absence of the injured Jordan Henderson and Adam Lallana.
Anfield's little magician finally rediscovered his wand. Coutinho was outstanding – this was the player who tormented defences back in the autumn prior to the ankle injury which derailed his season.
Klopp had warned the Blues that Coutinho had returned from international duty with a spring in his step after his midweek heroics against Paraguay but they were powerless to stop him from wreaking havoc.
His eighth goal of the campaign was a beauty and he then created the third for Origi before departing to thunderous applause.
If Coutinho was the star of the show then there was no shortage of highly impressive supporting roles.
No wonder Klopp reserved his biggest post-match embrace for Lucas Leiva.
The long-serving Brazilian midfielder, who got his first taste of the derby a decade ago, emphatically repaid the manager's faith in picking him as Lallana's replacement.
Lucas was immense as he thundered into tackles, won his aerial battles and kept Liverpool on the front foot. He epitomised the kind of controlled aggression that Klopp had demanded.
Lucas clearly relished every second of it - how glad he must be that he spurned Inter Milan's advances back in January. Whatever the future holds for him this summer, he richly deserves an Anfield testimonial.
Ahead of him, Emre Can and Gini Wijnaldum ran themselves into the ground as they pressed relentlessly and forced the Blues to cough up possession.
Behind him, centre-backs Dejan Lovren and Joel Matip did brilliantly to nullify the threat of Romelu Lukaku, who never looked like adding to his 22-goal haul.
Lovren hadn't graced the Premier League since late January due to his injury troubles and had fallen below Ragnar Klavan in the pecking order.
But the Croatian defender grasped his opportunity to shine with Matip typically ice cool alongside him.
There was no sign of tired limbs after the international break as the Reds quickly stamped their authority on the contest.
The hosts led inside eight minutes as Mane started where he left off in the previous derby with that dramatic late winner at Goodison back in December.
The £30million man burst goalwards after a clever one-two with the tireless Roberto Firmino.
Mane shrugged off the attentions of Tom Davies and got away from Ashley Williams before drilling low into the far corner. Anfield erupted.
Coutinho had helped create the space for Mane with his intelligent movement and the Brazilian's influence on the contest grew and grew. Belief was surging through his veins.
Liverpool's energised front three continued to prey on the inexperience of Pennington and the ageing limbs of Phil Jagielka.
Coutinho went close before the Reds conceded an equaliser against the run of play in infuriatingly familiar fashion.
Jagielka flicked on Leighton Baines' corner and Pennington slotted home his first goal for the Blues.
Liverpool's response was swift with Coutinho taking centre stage. He surged forward with real intent, turning away from Idrissa Gueye and Pennington before planting it expertly into the top corner.
Everton lost their heads. Barkley, who was lucky to avoid a card earlier for a horrible lunge on Can, then recklessly planted his studs on Lovren's ankle. Klopp was rightly incensed that Barkley escaped with a yellow.
There was a scare early in the second half when Ashley Williams got on the end of Barkley's free-kick but Simon Mignolet, who was rarely tested, clung on.
With Origi replacing the injured Mane, Liverpool continued to press in search of greater reward. It arrived on the hour mark.
It was in this fixture last season that Origi was cut down in the form of his life by Ramiro Funes Mori's shameful tackle. The young Belgian has been trying to get back to those heights ever since.
But he took a welcome leap in that direction with this lively cameo. Origi had only been on the pitch for three minutes when he latched on to Coutinho's pass and hammered his first league goal since December past Robles from the edge of the box.
There was much to admire about Origi's all-round display as he continued to put himself about.
Mignolet clawed away Mason Holgate's deflected effort before teenager Trent Alexander-Arnold was denied a dream goal by Robles' acrobatic save on his derby bow.
Klopp's slick Liverpool were worthy winners as Everton departed with that same old sinking feeling.
LIVERPOOL (4-3-3): Mignolet, Clyne, Lovren, Matip, Milner, Lucas, Can, Wijnaldum, Coutinho (Alexander-Arnold 73), Mane (Origi 57), Firmino (Klavan 90). Subs: Karius, Moreno, Grujic, Woodburn. BOOKING: Can. GOALS: Mane 8, Coutinho 31, Origi 60.
EVERTON (3-4-3): Robles, Pennington (Barry 67), Williams, Jagielka, Holgate, Gueye, Davies (Valencia 67), Baines, Barkley, Calvert-Lewin (Mirallas 82), Lukaku. Subs: Stekelenburg, Kenny, Lookman, Kone. BOOKINGS: Davies, Barkley, Williams. GOAL: Pennington 28.