Sanchez had previously been linked with a switch to Premier League leaders Manchester City, but the forward has said that he jumped at the opportunity to work with Jose Mourinho at Old Trafford.
The 29-year-old told the club's official website: "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.
"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 have not disclosed the length of Sanchez's contract, but reports have indicated that the player has penned a four-year deal.