Arsenal stunned Louis van Gaal’s side, who had been top before the weekend, by scoring three times in the first 20 minutes through Alexis Sanchez (two) and Mesut Ozil.
Manchester United dropped to third, two points behind leaders Manchester City who have 18 from eight games.
Earlier, Liverpool’s 1-1 draw at Everton could not save manager Brendan Rodgers from the sack, a decision announced three hours after the end of the game.
Romelu Lukaku equalised for the hosts after Danny Ings had headed Liverpool in front.
In Sunday’s other match, Tottenham Hotspur’s Christian Eriksen scored with two stunning free-kicks to earn his side a 2-2 draw at Swansea City.
Andre Ayew and an own goal by Harry Kane had twice put the Welsh side in front.
Spurs stayed eighth, level on 13 points with Everton and still three ahead of Swansea.