Jurgen Klopp's men are now nine points clear of Arsenal in sixth place, though have played three games more.
Liverpool's decision to hire a private jet to rush the Philippe Coutinho back from international duty in South American qualifying paid off, with the Brazilian scoring his team's second goal after a weaving run and also setting up Divock Origi for the clinching third in the 60th minute.
It's now 13 league games without defeat against Everton for Liverpool — equaling a club record — and an important win in the context of Champions League qualification, as the Reds consolidated its place in the top four.
Sadio Mane put Liverpool ahead in the eighth minute with another solo effort, before later going off with a left leg injury, and Matthew Pennington equalized against the run of play for an Everton side that came to Anfield in seventh place and as the league's form team.
Liverpool handled the occasion better than its rival.
It was a typically combustible match — there have been more red cards in the Merseyside derby (21) than any other Premier League fixture — and Everton midfielder Ross Barkley will count himself fortunate to finish the game. He avoided a booking for a bad early foul on Emre Can and then only got a yellow card for a crude, studs-first lunge on the left ankle of Dejan Lovren.
Barkley struggled for composure in a frenetic game, unlike Liverpool counterpart Coutinho, who oozed class and assuredness — not least when he sent Barkley skidding on his backside with a wonderful shimmy early in the second half.
Coutinho's goal was the standout moment, overshadowing Mane's goal as Everton paid for standing off the Senegal forward. Exchanging a one-two with Roberto Firmino, Mane shrugged off Tom Davies, drove past Ashley Williams and slid a low, angled shot into the far corner.
Everton equalized as Pennington marked his first appearance of the season for Everton by scoring his first-ever goal for the club, slotting home from close range after Phil Jagielka flicked on a corner at the near post.
Coutinho, a scorer for Brazil in midweek, then took center stage. He reclaimed the lead for Liverpool in spectacular style by picking up the ball 40 meters (yards) out, drifting past Idrissa Gueye, cutting inside Pennington and curling brilliantly into the top corner.
More skill from Coutinho led to the third goal, the Brazilian following a jinking run with a pass inside to Origi. The striker — on the field only three minutes as a substitute for Mane — took a touch and sent in a 30-meter shot that appeared to wrong-foot goalkeeper Joel Robles and flew into the net.
Everton's last league win over Liverpool was in October 2010.