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The Dutchman was unhappy at what he saw as an overreaction to challenges from his players.
"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," said Koeman.
"According to their bench, the referee should have shown eight red cards."
Koeman made it clear he was not referring to counterpart Jurgen Klopp, but some members of the Liverpool staff.
Liverpool-Everton games have seen 21 red cards in the Premier League era, more than any other fixture.
In a game where goals from Sadio Mane, Philippe Coutinho and Divock Origi took Liverpool third, four yellow cards were shown.
"Maybe one or two tackles were a little too much. From both sides I saw tackles that were maybe more than a yellow," added Koeman, whose side missed the chance to go sixth.
"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.
Liverpool boss Klopp did not respond directly to Koeman's comments, but was asked about a challenge by Ross Barkley and Dejan Lovren, where the Everton midfielder made heavy contact with home defender's ankle.
"I'm still a kind of a guest in this country," said the German. "How can I decide what is OK and what is not OK? My opinion is that the players should leave the pitch healthy and fit but not injured.
"It's not my job to judge it. If you saw something - say it. If not be quiet."
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Liverpool did suffer the loss of Mane early in the second half, the forward falling awkwardly after trying to win the ball from Leighton Baines.
The 24-year-old former Southampton man injured his left leg and is likely to miss the midweek visit of Bournemouth.
"When I saw him in the dressing room, it didn't look like he'll be ready for Wednesday," said Klopp.
"Obviously he has pain and it didn't look like it's only a knock or a muscle or something, but we have to wait.
"When I came in I was looking for him immediately and I saw him on the bed. He'd already had treatment, and it didn't look like he felt really comfortable."
Everton manager Ronald Koeman criticised some members of the Liverpool coaching staff following the Reds' 3-1 win at Anfield on Saturday.