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Liverpool had a chance on the break right at the start of the game, but Jose Enrique squandered his pass with Raheem Sterling well-placed.
Kevin Mirallas broke through the entire visiting defence in the ninth minute, but it was eventually blocked for a corner by Daniel Agger.
Five minutes later Liverpool were in front. Enrique found Suarez, whose shot took a huge deflection off Leighton Baines and went in for an own-goal.
He could not help but avoid controversy, however, as he ran to the Everton bench to dive in front of David Moyes following the Everton manager's recent comments.
In the 20th minute it got better for the Reds and this time it was Suarez's goal. Everton's defence went missing from a Steven Gerrard free kick and the Uruguayan glanced the ball home.
Suarez could have extended Liverpool's lead, but his fizzing low drive from 20 yards went just wide.
Instead, Everton drew level through Steven Naismith's first goal for the club. He was found by Marouane Fellaini inside the six-yard box to tap home.
Everton dominated the rest of the half but the sides went in level.