The 31-year-old did not get anywhere near the heights expected by fans after joining from Arsenal in January 2018 in a deal which saw Henrikh Mkhitaryan go the other way.
Having scored just five goals in 45 appearances, Sanchez was allowed to spend the 2019-20 season on loan at Inter Milan, a period in which United covered a hefty portion of his wages.
The Chile international has done enough in Serie A to secure a permanent move, bringing an end to an uninspiring stint in the north-west of England.
The Italian club said in a statement: “FC Internazionale Milano announce that Alexis Sanchez has joined the Nerazzurri on a free transfer.
“The Chilean forward has signed a permanent contract with the club until 30 June 2023.”