The result means that Liverpool suffered their fourth consecutive defeat in the league and fourth in a row at home.
Liverpool's dismal day was compounded when they lost captain Jordan Henderson to injury in the first half
Everton go level on points with Liverpool with both teams three points off the top four places, but Everton have a game in hand.
"I am very pleased for the club and the supporters," Everton boss Carlo Ancelotti told Sky Sports postmatch. "I hope for sure that they are going to celebrate tonight. It was a good performance. A lot of fight and spirit. I am really happy."
Everton took the lead inside three minutes when James Rodriguez played a through ball to Richarlison, who produced an emphatic finish.
The Liverpool captain then limped off just before the half-hour mark with an injury and was replaced with youngster Nathaniel Phillips.
Just before 70 minutes, Mohamed Salah was denied from close-range by another great Pickford save.
With eight minutes remaining, Dominic Calvert-Lewin was brought down by Alexander-Arnold inside the box and Sigurdsson converted the penalty to seal the win.
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Everton's long-serving skipper Coleman was delighted to have finally tasted success at Anfield.
"It's an amazing feeling, after coming here all those years and letting ourselves and more importantly letting the blue side of the city down," he said.
"It's been hard. We can try and brush it off but when you have lived in the city for 10 years it has been difficult to take.
"I'm fed up with coming out with the same cliches after losing. I'd thank the manager for the way he set us up and the players put in an unbelievable shift. That was against the champions, so we are delighted."
Speaking to Sky Sports after the match, Klopp said: "The first goal which we have to defend better gave the direction of the game. It was unnecessary.
"There were plenty of good football moments so we did well from a creative point of view but we had to chase the game and were not clam enough. Everton defended deep and full of passion but there were still moments where we were completely free in the box but we didn't finish."