The Chile international looked destined for Manchester City until recently, with the Sky Blues appearing set to wrap up a move for around £20m having originally agreed a £60m fee with Arsenal last summer.
But United stole in with a late approach to secure the signing of Sanchez, who is set to earn a Premier League record of around £350,000 a week, on a four-and-a-half-year deal having allowed his Arsenal contract to run down to its final six months.
“I am thrilled to be joining the biggest club in the world," Sanchez, who takes the club's No.7 shirt, told Man Utd's official website. "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.”
Man Utd manager Jose Mourinho - who got one over his old rival Pep Guardiola at Man City with the signing - praised his new player's impressive attributes and thanked Mkhitaryan for his contribution at Old Trafford.
“Alexis is one of the best attacking players in the world and he will complete our very young and talented group of attacking players," the Portuguese said.
"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, who successfully sorted out his visa on Monday afternoon along with Sanchez, said it was a "dream" to play for the Gunners.
"I’m very happy that we could finish this deal and I’m very happy to be here," the attacker told Arsenal's official website.
"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 coach 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."
The arrival of Sanchez will be considered a real coup by United after the player had come so close to signing with rivals City, who finally pulled the plug on their own interest in the 29-year-old last Sunday.
Sanchez spent three-and-a-half years at Arsenal, having arrived at the Emirates Stadium from Barcelona in 2014 for around £32m. He previously had spells with Cobreloa, River Plate, Colo Colo and Udinese.
The completion of the transfer comes almost a decade after United's first serious interest in the Chilean while still playing with Udinese in Serie A, and is the first deal to see an Arsenal player leave for Old Trafford since Robin van Persie's high-profile move in 2012.
Mkhitaryan's exit as part of the deal ends the Armenian's 18-month spell at rollercoaster Old Trafford since arriving from Borussia Dortmund in the summer of 2016, with the 29-year-old signing with the Gunners until the summer of 2021.
He spent two lengthy periods out of favour under Mourinho but did play a huge part in United's Europa League triumph in 2016-17, scoring the second goal in the 2-0 victory over Ajax in last May's final.