The 27-year-old, who was instrumental in helping Milan return to the Champions League for the first time since 2014 last season as he contributed four goals and 10 assists, has signed until 2024.
"It's a great thing, we know that Inter are a great team, a great club with a nice atmosphere, and great fans," Calhanoglu said. "I'm very happy to be here, I'm going on vacation now, but I already have good feelings. We want to win the Scudetto this year too, so we'll see."
Calhanoglu added that he had already spoken with new Inter coach Simone Inzaghi.
"Yes, we have already spoken," he said. "I know his game system, the 3-5-2 which can change into a 3-4-1-2 depending on the situation. I really like how he spoke; I have heard many positive things. There are great objectives, which is important because I also want to win something in Italy."
Milan had hoped to keep the playmaker who joined from Bayer Leverkusen in 2017, but were unable to agree a contract extension.