It took just 32 minutes for Everton to pierce the City defence when Leon Osman bent a shot into the net.
Despite Steven Pienaar being sent off on the hour, City just could not take advantage.
Then in stoppage time, Nikica Jelavic secured the win by completing a speedy counter attack with Everton's second goal.
The result leaves City 12 points behind United ahead of the Reds' match against Reading later today.
