At times, emotions in the game threatened to spill over, with the players clashing en masse after Liverpool did not put the ball down with Richarlison on the ground in the first half.
Anthony Gordon twice went down in the box following home challenges, with the first prompting the referee to book him for diving.
Everton boss Frank Lampard felt both should have resulted in a penalty, which would have given the away side the chance to open the scoring.
"I think they both could have been penalties but you often don't get them at Anfield," said the Toffees boss, whose side are now in the bottom three following the defeat, coupled with Burnley's win over Wolves on Sunday.
"The fact there's contact and he gets booked is crazy. The second one was a foul [anywhere else] on the pitch. Sometimes you don't get them here."
Reds boss Jurgen Klopp was adamant neither were spot-kicks, saying: "It [the second] was clearly no penalty. He's really good but he's lucky not to get a second yellow. The first was not a penalty and the second wasn't."
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