The striker left Manchester United officially on Wednesday for the Nerazzurri.
"Hello there Inter fans. I am very, very happy to be staying here with you," said Sanchez in a video message posted on the official Inter Twitter feed.
“I think that I had this desire to stay here because I have found a family, in the staff and everyone who is eager to fight and win something for this club.
“I am hungry and want to win something, so that is why I am so happy to be here, with you, to win something."