Sources had told ESPN FC in May that Godin had signed a three-year contract worth €6.75 million at San Siro, and the Nerazzurri have confirmed the news on July 1, the day the Italian transfer window opened.
The 33-year-old becomes new Inter boss Antonio Conte's first signing since being appointed as Luciano Spalletti's replacement.
Godin called a news conference to announce his departure from Atletico, which was attended by many of his teammates as well as manager Diego Simeone, but refused to elaborate on his next destination.
Following the end of the La Liga season, he joined up with Uruguay for the Copa America, where La Celeste were beaten on penalties by Peru in the quarterfinals.
Inter qualified for the Champions League on the final day of last season, but finished 21 points behind runaway leaders Juventus in Serie A.