Sadio Mane's opener was cancelled out by Matthew Pennington's first Everton goal, but strikes from Coutinho and substitute Divock Origi confirmed Liverpool's league double over their neighbours.
Liverpool had not lost in their last 12 league derbies and took the lead in the eighth minute when Sadio Mane exploited poor Everton defending to net his 13th goal of the season, playing a one-two with Roberto Firmino and holding off two opponents before firing low past Joel Robles.
Everton were without injured defenders Ramiro Funes Mori and Seamus Coleman their unfamiliar back line struggled to cope with Liverpool's attackers.
But Everton's Pennington marked his first appearance of the season with an equaliser against the run of play in the 28th minute, prodding home from a corner after Dejan Lovren's clearance.
The inexperienced defender was caught out less than three minutes later when Coutinho tore past Idrissa Gueye, cut in past Pennington and slammed the ball into the far corner.
Anfield paid tribute to late Liverpool coach Ronnie Moran before the match and the emotion of the occasion carried over with Everton's Ross Barkley lucky to escape a red card for a high tackle on Lovren in the 39th minute.
Mane was injured after a collision with Leighton Baines, but his replacement Origi scored three minutes after coming on with a thumping strike from outside the area after being played in by the outstanding Coutinho.
"The boys did everything," Liverpool manager Juergen Klopp told Sky Sports after extending his 100 percent record in Merseyside derbies. "It's a deserved win, we played good football and we coped well."
Everton's last league win at Anfield came in 1999, and Ronald Koeman's side showed little sign of finding a way back into the match, with 21-goal top scorer Romelu Lukaku cutting an isolated and anonymous figure.
"I'm am really proud about the team, we lost but the final result is not reflecting of our performance," Koeman said. "We conceded too easy the second and third."
The win took Liverpool up to 59 points, level with Tottenham Hotspur ahead of their match at Burnley later on Saturday and two points clear of Manchester City who are at Arsenal on Sunday.
Everton, who travel to Manchester United on Tuesday, remained in seventh on 50 points.
(Reporting by Ed Dove, editing by Ed Osmond)
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