Divock Origi and Xherdan Shaqiri both came in from the cold and netted in a pulsating start to the action.
Michael Keane reduced the arrears for the visitors before Origi and the magnificent Sadio Mane put the hosts firmly in control. Richarlison then pulled another one back right at the death of an incredible first half.
The second period was not quite as thrilling but Liverpool produced a more controlled performance, with Georginio Wijnaldum putting the icing on the cake late on to ensure they remain eight points clear at the summit.
The victory also saw Klopp become the second fastest manager to clock up 100 wins in the Premier League and means the Reds have set a new club record of 32 top-flight games unbeaten.
In contrast, Everton's barren run at Anfield stretches to 20 years and counting, with Marco Silva's position appearing to be somewhat untenable as the Blues dropped into the relegation zone.
Liverpool march on, but it looks bleak for Silva.
The records pile up for Liverpool and Klopp as they continue to rack up the wins. This was a superb performance where fringe players came in and delivered alongside some of the established stars. After the craziness of the first period, the Reds controlled the second to see out an impressive triumph. The one minor frustration for the German will be that elusive clean sheet (13 games) as some sloppy defending allowed Everton to stay in touch for the majority of the contest. The Reds have got a hectic December schedule off to the best possible start, but that's far from the case for Everton. The Blues have now lost eight of their last 11 league matches and have dropped to 18th in the table. Silva needed a positive result at the home of his club's great rivals, but got a hiding instead. The teams will meet again in the third round of the FA Cup in January, but it's looking unlikely Silva will be in charge then unless there is a very dramatic turnaround in fortunes.
MAN OF THE MATCH
Sadio Mane (Liverpool). Stole the show with a pair of sumptuous assists and a superb goal of his own. Showed just why he has had the likes of Lionel Messi hailing his brilliance, but should have added to his tally when through on two occasions late in the match.
LIVERPOOL: Adrian 6, Alexander Arnold 8, Van Dijk 7, Lovren 8, Robertson 7, Milner 7, Wijnaldum 8, Lallana 8, Shaqiri 7, Origi 8, Mane 9. Subs: Henderson 7, Firmino 7, Gomez n/a.
6' - GOAL! Liverpool 1-0 Everton. Origi latches on to Mane's clever through ball, races beyond Pickford and coolly rolls the ball home.
17' - GOAL! Liverpool 2-0 Everton. Shaqiri slides the ball into the far corner after making a smart diagonal run on to Mane's pass. TAA had set the Reds in motion with a sensational cross-field ball.
21' - GOAL! Liverpool 2-1 Everton. Keane gobbles up a chance from close range after getting the better of Lovren as he attacked Iwobi's cross.
32' - GOAL! Liverpool 3-1 Everton. Origi takes Lovren's pass down brilliantly before lobbing Pickford with a wonderful finish.
45' - GOAL! Liverpool 4-1 Everton. Liverpool break at speed. TAA feeds Mane and he wastes little time sweeping a first-time finish into the bottom corner from the edge of the box.
45+3' - GOAL! Liverpool 4-2 Everton. Richarlison guides Bernard's cross into the far corner. It's raining goals at Anfield.
90' - GOAL! Liverpool 5-2 Everton. It's five! Wijnaldum puts the icing on the cake with a low finish into the far corner after wonderful skill from Firmino on the left.
Origi has now scored in all four matches he's played against Everton at Anfield.
Mane assisted twice in a Premier League game for the first time since March 2018 v Watford.
Liverpool’s victory over Everton was Jürgen Klopp’s 100th Premier League win in his 159th match in charge in the competition; among all managers, only José Mourinho (142 matches) enjoyed 100 wins in fewer games in the competition’s history.
Since the start of last season, Trent Alexander-Arnold has provided more Premier League assists than any other player (18), with 15 of those coming in 2019.
Everton have conceded five goals against Liverpool in a match for the first time since November 1982, when they lost 5-0 under Howard Kendall against a Reds side managed by Bob Paisley.
Everton find themselves in the relegation zone after playing at least 15 Premier League matches for the first time since April 10th 1999 (after 32 games).