United handed Sanchez a £9million pay-off as part of his free transfer to Inter after finally ending his nightmare spell at Old Trafford on Thursday, where he scored just five goals and produced nine assists in 45 matches.
The Premier League club also waived a transfer fee so that Inter could turn Sanchez's loan at the San Siro into a permanent move and sign the Chile star on a three-year deal worth £130,000-a-week.
"I'm really happy to be staying here with you, Inter fans," Sanchez told Inter's social media channels. "I've found a family here, and the staff are great.
"[It's] a group where there's a real desire to win something big. People here want to fight together as a team to give the club the success it deserves.
"I'm hungry and I want to win - my goal is to make all our fans happy. I'm here to win, and I'm happy that I'm staying here at Inter!"