Klopp's side upped the pressure on their top-four rivals - moving level with second-placed Spurs after their early kick-off.
But a potentially serious injury to Mane could overshadow their derby victory moving into the run-in.
FACTS, STATS, GOALS AND LOLS
Jurgen Klopp showed his class with a heartfelt message to stricken Everton star Seamus Coleman in the programme – highlighting that some issues transcend sporting rivalries.
Sadio Mane opened the scoring with a brilliant finish despite being off balance. The Senegal flier has been a brilliant signing.
12
Sadio Mane appeared to seriously injure his knee in a worrying sight for Liverpool fansCredit: Rex Features
12
Matthew Pennington prods home after Liverpool failed to defend a cornerCredit: Reuters
Everton fielded kids all over the park and at times their lack of experience showed. Exposing them to matches like this will benefit the Toffees in the future.
Liverpool’s movement was too much for their rivals. The Reds’ attacking talents found space between the lines time and time again.
Liverpool’s defending proved to be their Achilles heel again. Until the frailty at the back is sorted they will struggle for titles.
Philippe Coutinho showed no signs of jet lag as he curled in an absolute peach to restore Liverpool’s lead.
Divock Origi sealed the win after coming on for the injured Mane. But the Senegal star will be a huge miss for Klopp's side.
12
Philippe Coutinho restored Liverpool's lead with a brilliant secondCredit: Reuters
12
Brazil star Coutinho curls home for 2-1Credit: Reuters
12
Jurgen Klopp was as animated as always on the touchlineCredit: Reuters
12
Ross Barkley goes in late on Dejan LovrenCredit: Rex Features
12
Mane celebrates his opener in front of the Liverpool bench
SOCIAL SAYS
BIG PICTURE
12
WHAT THEY SAID
Dejan Lovren on Ross Barkley's challenge: "Listen, I don't want to talk about this now. You can expect in the derbies tackles like this. Tomorrow there will be headlines about this. I'll leave it to the manager."
Divock Origi: "The team are really strong, and we have to finish strong. I try to do what I have to do and enjoy the game. It was nice to score today.
"It is special, a very special game, it was very big. We are going to enjoy it and learn from it, there’s some good things, some less good things, we have another game in the week."
12
Sadio Mane gets Liverpool off to a flier with a tidy finishCredit: Rex Features
12
Tom Davies went in the book for a first-half lungeCredit: Getty Images
12
Liverpool fans pay tribute to Ronnie Moran ahead of kick-offCredit: Reuters
DREAM TEAM RATINGS
LIVERPOOL: Mignolet 6, Clyne 6, Lovren 7, Matip 7, Milner 7, Lucas 8, Can 6, Wijnaldum 6, Mane 8, Firmino 7, COUTINHO 8. Subs: Klavan 6, Alexander-Arnold 6, Origi 7.
EVERTON: Robles 5, Holgate 7, Williams 6, Jagielka 8, Pennington 6, Gueye 7, Davies 6, Baines 6, Barkley 7, Calvert-Lewin 6, Lukaku 6.Subs: Barry 6, Mirallas 6, Valencia 6.