Mkhitaryan, 29, has signed a long-term contract to become the Gunners' second January signing, following Konstantinos Mavropanos from PAS Giannina.
“Henrikh is a very complete player," manager Arsene Wenger told Arsenal's official website.
"He creates chances, he defends well, he can absorb distances and he’s very committed as well. He’s a player who has all the attributes.”
As part of the transfer, Sanchez will join Jose Mourinho's United.
Both players could make their PL debuts next week, when Arsenal travel to Swansea City on 30 January, and Manchester United go to Tottenham Hotspur the next evening.