An early Sadio Mane goal put Liverpool ahead only for Matthew Pennington to peg the hosts back 20 minutes later.
Three minutes on and Liverpool had restored their lead through Philippe Coutinho. The Brazilian used his dazzling feet to work some room before curling in a beauty into the top corner.
Divock Origi killed the game in the second half when he scored the Reds’ third on the hour mark.
As expected with a Merseyside derby, big tackles came flying in from both sides, but Everton’s Ross Barkley arguably could have seen red in the first half after a stud-high lunge on Dejan Lovren.
Speaking after the game, Koeman hit out at Klopp for the German’s touchline decorum.
Speaking to Sky Sports, he said: “Tackles are part of football – I saw some tackles from Lucas.
“It’s all about football but we don’t make a show from the bench like Liverpool. It’s football – a hard and fair game.
“I don’t like coaches from the bench the whole time shouting to referees and linesmen, making a big show about tackles. They were crazy. They didn’t need the physio on the pitch. It’s a man’s sport.”
It’s the second time in 24 hours Koeman has hit out at a fellow football manager.
The Everton boss made his feelings about Martin O’Neill clear via Twitter on Friday.