"I am thrilled to be joining the biggest club in the world," said 29-year-old Chile international forward Sanchez.
"I have spent three-and-a-half wonderful years at Arsenal and I bring with me very positive memories of that great club and its fans.
"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 is expected to be the highest ever paid player in the Premier League, with British media reporting a pre-tax salary of 500,000 ($695,000, 567,000 euros) a week.
The swap deal that sees Armenia international winger Mkhitaryan, also 29, head to the Emirates following a mixed 18 months at Old Trafford had been expected for several days.
Sanchez had joined Arsenal in 2014 for 31.7 million after three years at Barcelona.
"Alexis is one of the best attacking players in the world and he will complete our very young and talented group of attacking players," said United manager Jose Mourinho.
"He will bring his ambition, drive and personality, qualities that make a Manchester United player and a player that makes the team stronger and the supporters proud of their club dimension and prestige.
"I would like to wish Henrikh all the success and happiness that I am sure he is going to get. He is a player that we will not forget, especially for his contribution to our Europa League victory."
Mkhitaryan joined United for around ?30 million in 2016 after three years at Borussia Dortmund.
Arsenal also published the news of the swap deal on their website."I'm very happy that we could finish this deal and I'm very happy to be here," said Mkhitaryan.
"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."
Arsenal boss Arsene Wenger added: "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."...