It was aimed as fut coins a rebuke to Moyes, who had criticised the Uruguayan for diving in the build-up to the game, but Everton reacted quickly and drew level before half-time through goals from Leon Osman and Steven Naismith.
Suarez thought he had had the final say after poking the ball home in the fourth minute of injury time, but the goal was ruled out in what a looked a very tight call indeed.
"The goal at the end was clearly a goal and we are disappointed not to get it," said Liverpool coach Brendan Rodgers. "I think the linesman gave it but everyone in the ground thought it was a goal. That should have been a goal and an outstanding hat-trick for Luis Suarez."