"I am thrilled to be joining the biggest club in the world," said Sanchez, who has penned a four-and-a-half-year deal with the Red Devils worth £14m a year after tax.
"The chance to play in this historic stadium and to work with Jose Mourinho was something I could not turn down.
"I am very proud to be the first Chilean player ever to play for United's first team, and I hope I can show our fans all around the world why the club wanted to bring me here."
Sanchez, who was Arsenal's top scorer in all competitions last season with 30 goals, scored 80 times for Arsenal since joining from Barcelona for £30m in July 2014.
Mkhitaryan has gone in the other direction, signing a long-term contract with the Gunners.
He netted 13 goals in 63 appearances for United, having moved to Old Trafford from Borussia Dortmund for £26.3m in July 2016.
“I’m very happy that we could finish this deal and I’m very happy to be here. It’s a dream come true because I’ve always dreamed of playing for Arsenal. Now that I’m here, I’ll do my best for this club to create history," the Armenian told Arsenal's website.
Gunners manager Arsene Wenger was also delighted to bring in a player of Mkhitaryan's quality.
"Henrikh is a very complete player. He creates chances, he defends well, he can absorb distances and he’s very committed as well. I must say he’s a player who has all the attributes," Wenger added.