Javier Hernandez celebrates after scoring United's second goal
Darron Gibson, who left Old Trafford earlier this month, struck on the hour for Everton against City, which is now only top due to its superior goal difference over United.
Defending champion United scored both goals from the penalty spot through Javier Hernandez and Dimitar Berbatov.
Third-place Tottenham swept past Wigan 3-1, moving seven points clear of Chelsea, which was held to a 1-1 draw at Swansea.
Liverpool provisionally climbed above Arsenal into fifth place with a 3-0 win at Wolverhampton Wanderers.
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