Jose Mourinho’s side took the lead after just four minutes when Antonio Valencia took advantage of a poor Laurent Koscielny pass to slot an opening goal past Petr Cech.
The visitors doubled down on their fast start when Lingard took the ball off Shokdran Mustafi, linked up superbly with Romelu Lukaku and Anthony Martial before finishing the move off himself on 11 minutes.
An end-to-end first-half saw Arsenal create plenty of chances, but fail to beat an inspired David de Gea in the opposition goal.
Arsene Wenger’s men did finally find a way past the Spaniard when Alexandre Lacazette scored shortly after the break, but Lingard scored a vital third just after the hour mark.
Paul Pogba was handed a straight red card after 74 minutes for a studs-up challenge on Hector Bellerin, but the visitors held on for the win in a thrilling contest.
The Pogba red was the only blot on the Red Devils’ copybook, as the Frenchman will now miss next weekend’s Manchester derby.