A Wayne Rooney penalty and an Abou Diaby own goal gave United a 2-1 win.
Arsenal manager Arsene Wenger's frustrations boiled over in stoppage time when he kicked a water bottle after a tying goal was ruled offside.
Wenger was subsequently sent off by referee Mike Dean on the advice of fourth official Lee Probert that was called an error and will result in an apology, but League Managers Association chief executive Richard Bevan said it was a mistake.
Bevan said: "Although correct in 'law,' it was completely out of context in the game and it was followed by the nonsense which followed over where Arsene Wenger should sit. I've spoken to [Premier League referees chief] Keith Hackett and he fully recognizes the situation was an error and an apology will follow to Arsene Wenger.
"Lee Probert totally failed to manage the situation and created a needless pressure point taking the focus away from the pitch in a big event with only a minute to go."