The visiting Toffees dominated large parts of the opening half and went on to break the deadlock in the 33rd minute. Full-back Tony Hibbert sent over a cross from the right which Nikica Jelavic managed to guide over United goalkeeper David de Gea with a looping header.
Everton's goal sparked a reaction from the hosts, who began to exert pressure on their opponents.
United began the second half in the ascendancy and took the lead for the first time in the 57th minute. Danny Welbeck collected a knock down from Nani before curling the ball into the top corner of Tim Howard's net from the edge of the area.
Nani turned goalscorer three minutes later to put United 3-1 up when he lifted the ball over Howard after being put through by Welbeck.
David Moyes's visitors refused to accept defeat though and reduced the arrears in the 67th minute courtesy of Marouane Fellaini, who volleyed in a cross from Hibbert.
However, United opened up a two-goal margin again after 69 minutes thanks to Rooney, who guided the ball past Howard following a cross from strike partner Welbeck.
Everton responded once again as Jelavic claimed his second goal of the afternoon in the 83rd minute with a strike from close range.
Then, two minutes later the away side levelled up proceedings when Steven Pienaar slid in to connect with a low cross from Fellaini to beat De Gea from six yards out.
United poured forward in search of a winning goal in the remaining minutes, but they were unable to find the net for a fifth time.
Their lead at the top will be cut to three points if Manchester City defeat Wolverhampton Wanderers later today.