Sanchez has signed a four-and-a-half year contract at Old Trafford worth reportedly around £500,000-a-week while Mkhitaryan has penned a long-term contract with Arsene Wenger's side, who will now press on with the signing of Borussia Dortmund forward Pierre-Emerick Aubameyang.
Mkhitaryan's agent Mino Raiola said last week that a deal to bring Sanchez to United would not materialise if the Armenian did not want to take his place in north London but the 28-year-old's move to the capital is now complete, though regulatory processes have not yet finished according to Arsenal's official website.
Sanchez was widely expected to join United's arch-rivals Manchester City but Pep Guardiola's side refused to bow to the former Barcelona star's demands as they feared the dressing room harmony would have been damaged by making the 29-year-old their highest-paid player. United had no such qualms, though, and Sanchez is keen to show the club's supporters what all the fuss is about.
"I am thrilled to be joining the biggest club in the world. 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," Sanchez told United's official website.
"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."
United boss Jose Mourinho also added his two cents over Sanchez. The Portuguese lauded his side's capture of one of the best attackers in world football and also paid tribute to the departing Mkhitaryan, who endured a frosty relationship with the former Chelsea boss.
"Alexis is one of the best attacking players in the world and he will complete our very young and talented group of attacking players," Mourinho 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."
Meanwhile, Mkhitaryan expressed his happiness over becoming an Arsenal player: "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."
While Arsenal promise to be very busy in the final stages of the January window, United are not expected to bring in further additions. Young defender Axel Tuanzebe is expected to leave Old Trafford on loan before the end of the month, with Aston Villa keen to push through a deal to bring him to the Midlands this week, according to the Telegraph.