Liverpool added to the growing pressure on Everton manager Roberto Martinez with a 4-0 victory which was as embarrassing for the visitors as it was comfortable for the hosts in the 226th Merseyside derby.
Defender Ramiro Funes Mori also added to the Spaniard's woes ahead of Saturday's FA Cup semi-final against Manchester United with this fixture's 21st red card in the last 47 Premier League encounters.
That was compounded by fellow centre-back John Stones limping off injured after the third goal went in. That was Daniel Sturridge's 50th goal for the club and followed two headers from Divock Origi and Mamadou Sakho just before half-time.
Philippe Coutinho's 76th-minute strike sent visiting fans heading for the exits and every 'Ole' which rang around Anfield towards the end was a dagger to the heart of Martinez and those who remained in the away end.Tweets from https://twitter.com/BelTel/lists/liverpool-v-everton
Sturridge, only on because Origi was carried off on a stretcher with a serious-looking ankle injury as a result of Funes Mori's stamp, became the fourth-fastest Liverpool player to his half-century by reaching the landmark in just his 87th match.
The England international's current strike-rate is better than that of Luis Suarez (91 games), John Aldridge (92), Michael Owen (93), Robbie Fowler (94), Kenny Dalglish (99) and Ian St John (111) - impressive considering the amount of time he has lost through numerous injuries.
By contrast Everton's premier striker Romelu Lukaku, included in Martinez's strongest possible line-up despite a looming Wembley date, barely had a sniff all night.
For 25 minutes Martinez's boldness appeared to be paying off but familiar weaknesses were soon exposed.
The visitors, whose longest winless run at Anfield was extended to 16 matches, were initially brighter and quicker to the ball, but unlike Borussia Dortmund a week ago they failed to take advantage of their early pressure.
Kevin Mirallas twice missed the target from good openings before Lukaku was denied by Sakho at full stretch.
Liverpool's best player during Everton's period of dominance, Adam Lallana, could curiously have had a hat-trick in the first half-hour, squandering the best chance as early as the sixth minute when he sprung the offside trap but shot straight at Joel Robles.
Everton's Gareth Barry, equalling David James' record of 571 Premier League starts, and James McCarthy appeared to have control of midfield but gradually that was wrested from them as Liverpool's relentless attacking started to take its toll.
Coutinho, who gave makeshift right-back Bryan Oviedo numerous problems, had two efforts from long distance while Brazilian compatriot Roberto Firmino was denied by Robles.
Everton's resistance finally gave way a minute before the break when Milner's hanging cross from the right to the far post saw Origi out-jump John Stones to head in his fifth goal in as many games, off his shoulder.
The Toffees' weakness has been to concede goals quickly, verging on a collapse at times, and this was exploited as Milner, this time from the left, exchanged passes with Lallana before crossing for Sakho, criminally unmarked, to power home the header.
Just when Martinez thought it could not get any worse after the break, along came Funes Mori's stamp on Origi. It resulted in the striker being carried off on a stretcher holding his head in his hands - which was also Klopp's reaction - and rightly meant the Everton man was dismissed.
The sight of the Argentina international kissing the badge on his shirt as he walked off was somewhat crass, especially with him now weakening Everton's defensive options even further for the semi-final.
Sturridge clinically dispatched Lucas Leiva's pass for the third on the hour before Stones' departure - which left midfielders McCarthy and Muhamed Besic in central defence - and Coutinho's smart finish compounded Everton's misery, a feeling experienced at Anfield all too often since 1999.
Liverpool manager Jurgen Klopp is not a man to hold back his emotions on the touchline but he expects his players to show cool heads in his first Merseyside derby.
The fixture has had the most red cards of any among top-flight teams - 20 in the last 46 - although none in the last nine.
And while Klopp may inspire his team, and the crowd, with his arm-waving, fist-pumping actions, he expects a more controlled approach on the pitch.
"My Merseyside derby experience will 100 per cent not be the reason for winning or losing," he said. "My job is to prepare the team. Before I came to Dortmund there were a lot of red cards in derbies (against Schalke). I don't like that showing.
"Maybe 20 fans want to see you do things like that but it will never help. A red card can happen if you are a second too late but not because you want to show 'I am the man'.
"I understand aggressiveness only one way and that's being prepared to hurt yourself and not somebody else.
"The best football is always full of emotion, passion and aggressiveness; legal aggressiveness. To handle emotions and the pressure, that's one of the big challenges in top-class football.
"We played Manchester United, I don't know if Everton is the biggest game or United is, but we handled it well. That's what I expect from this game too - that everybody sees from the first second on that we want to win this game. But with football."
Klopp arrived in October after predecessor Brendan Rodgers was sacked following a 1-1 draw against Everton at Goodison Park. In the intervening six months he has immersed himself in the city and feels he has a handle on what this match means, even if he cannot always understand the local fans.
"The Scouse language is still pretty difficult for me," he added. "But I know about derbies. It's my first derby with the team against a team in the same city, that's different."
For Everton, who have not won at Anfield since September 1999, it is their biggest week of the season with this match followed by an FA Cup semi-final against United on Saturday.
"I know about Everton and they're a big club, successful in the past, but I don't know what the situation is with the semi-final coming up," he said. "They are not used to, at the end of the season, playing two games a week too often.
"But we should think only about our game, our line-up and our situation."