Luis Suarez made one goal and scored another to put the visitors 2-0 up inside 20 minutes, celebrating his role in the opener with a gleeful dive right in front of the Everton dug-out.
It was aimed as 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 controversially ruled out despite the Liverpool striker being onside.
"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."
The result took Everton into the top four, bumping Arsenal down to fifth place, while Liverpool lie six points below their city rivals in 11th.
Liverpool struck first at Goodison Park, Leighton Baines inadvertently deflecting the ball into his own net in the 14th minute after Suarez drilled Jose Enrique's low cross back towards goal.
Suarez celebrated with a comical dive in front of the Everton bench and moments later he was celebrating again -- albeit in less provocative fashion -- after glancing Steven Gerrard's free-kick past Tim Howard.
Everton hit back within two minutes, however, with Osman drilling into the bottom-left corner after Liverpool goalkeeper Brad Jones failed to get enough purchase on a punch from an Everton corner.
The hosts' tails were up, and they equalised 10 minutes from the break when Naismith slammed home Marouane Fellaini's centre.
The weekend's headline clash takes place later on Sunday, when leaders Chelsea tackle third-place Manchester United at Stamford Bridge.
Chelsea have dropped just two points in eight league games this season but lost 2-1 at Shakhtar Donetsk in the Champions League in mid-week, while victory for United will take them to within a point of top spot.
Beforehand, West Bromwich Albion will bid to continue their impressive start to the season when they visit Newcastle United, and Tottenham Hotspur travel to second-bottom Southampton.
Source: AFP
Source: AFP