Sadio Mane, who scored a last-minute winner at Goodison Park in the reverse fixture, struck early on with his 13th Premier League goal of the season.
Defender Matthew Pennington, playing in his first Everton match of the season, responded for the visitors just before the half-hour mark but Philippe Coutinho promptly restored the hosts' lead with a wonderful finish.
Liverpool lost Mane to injury after the break, although that did not damage Jurgen Klopp's men, Divock Origi striking within four minutes of coming on to replace him to seal a crucial victory and star man Coutinho providing the assist.
YESSSSS! pic.twitter.com/LrhyuKmpa7— Liverpool FC (@LFC) April 1, 2017
The win sees Liverpool move above Manchester City and into third place after their fourth straight top-flight home victory.
They remain unbeaten against city rivals Everton since 2010 and have not suffered defeat in an Anfield derby since 1999.
Klopp has become the first Liverpool manager to win three straight league derbies, with the Reds moving nine points clear of Everton - who suffered just their second league loss of the calendar year - in seventh spot.
It took only eight minutes for the first goal to arrive when Mane exchanged passes with Roberto Firmino, surged past Ashley Williams and sent a left-footed shot through the legs of defender Pennington and into the far corner.
18 - Sadio Mane has now been involved in 18 league goals this season (13 goals, 5 assists), his best ever return in a PL campaign. Star.— OptaJoe (@OptaJoe) April 1, 2017
Pennington was playing as Everton were forced to re-shuffle their pack, deploying a three-man backline in the absence of injured defenders Seamus Coleman and Ramiro Funes Mori.
And Liverpool almost punished their weakened rivals with a second goal - Coutinho denied after collecting Emre Can's pass, cutting inside from the left and firing a strike at goal which Joel Robles tipped into the air, Phil Jagielka racing back to head clear near the goal-line.
Everton got back on level terms after 28 minutes with their first shot on target. Leighton Baines' corner from the left was flicked on by Jagielka and not cleared by Dejan Lovren – who was under pressure from Williams - allowing Pennington to apply the finish from inside the six-yard box.
But Liverpool hit back within just 177 seconds, Coutinho collecting Lucas Leiva's pass before outpacing Idrissa Gueye and taking advantage of half a yard given to him by Pennington to curl a sumptuous finish into the top corner, his sixth league goal of the season.
Despite protests from the hosts, Ross Barkley was only booked for a heavy challenge on Lovren's ankle as Everton went into the break behind.
Liverpool suffered a major blow early in the second half, losing the injured Mane after an awkward tangle with Baines, but his replacement Origi promptly struck on the hour mark.
After Simon Mignolet had kept out efforts from Williams and Barkley, Coutinho played a precise pass that allowed Origi to shoot first-time and blast past a motionless Joel from 20 yards, the Belgium international's first league goal since December.
Koeman brought on Enner Valencia and Gareth Barry, but Mason Holgate was the next player to threaten, having a deflected effort saved by Mignolet and heading off target from Baines' cross a few moments later.
At the other end, Joel was at full stretch to keep out a strike from substitute Trent Alexander-Arnold as Liverpool comfortably held on for victory on a frustrating day for Everton, whose in-form striker Romelu Lukaku did not manage a shot on goal.
Key Opta stats:
- Jurgen Klopp is the first Liverpool manager to win his first three league Merseyside derbies against Everton.
- Liverpool are unbeaten in their last 13 league encounters with Everton (W6 D7 L0), their joint-longest league run over their neighbours.
- Everton's last Premier League win at Anfield was in September 1999 (W0 D8 L9 since).
- Sadio Mane has scored in both Merseyside derbies this season, the first player to do that since Luis Suarez and Daniel Sturridge in 2013/14.
- Mane has now been involved in 18 league goals this season (13 goals, five assists), his best ever return in a Premier League campaign.
- Matthew Pennington became the fifth player to score his first ever Premier League goal in a Merseyside derby (Duncan Ferguson, John Arne Riise, Raul Meireles, Steven Naismith).
- Philippe Coutinho has assisted six Premier League goals this season, his second best tally in a Premier League campaign (seven in 2013-14).
- Only Olivier Giroud (seven) has scored more goals as a sub than Divock Origi (six) since the Belgian's Premier League debut in September 2015.