Nicolas Pepe gave the Gunners the lead after eight minutes, before centre-back Sokratis Papastathopoulos added a second strike shortly before half-time.
Andreas Pereira came closest to pulling one back for United shortly after being introduced in the second half, but Ole Gunnar Solskjaer's side couldn't find a way past Gunners goalkeeper Bernd Leno.
Watch the highlights from the game in the player above, and check out post-match reaction from the manager, as well as captain Harry Maguire, below.
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