The Nerazzurri have not won a trophy since 2011 as Juventus have enjoyed an incredible period of dominance in Italy.
The 33-year-old Uruguay international spent nine years at Atletico, helping them to LaLiga glory in 2013-14 and a pair of Europa League titles.
"I chose Inter because it's such a massive club, a great team with wonderful history and ambitions and desire to achieve big things," Godin told the club's official website.
"I'll try to do my bit and give everything I can both on and off the pitch.
"I want to help my team-mates and help the club to achieve big things. I chose Inter because they have an ambitious sporting project.
"They want to win titles again and they're a massive club in Europe and around the world. I've joined Inter to win, to give everything, to put my body and soul into fighting for this club both on and off the pitch."
🎙️ | INTERVIEW— Inter (@Inter_en) July 1, 2019
"I will be looking to share my experience both on and off the pitch" ⚫🔵🇺🇾@diegogodin’s first comments as a Nerazzurri player, exclusively to @Inter_TV #WelcomeDiego pic.twitter.com/e9U7Gz71KG
Godin also said Inter's Uruguay midfielder Matias Vecino played a role in him joining.
"He told me great things about the city and the team and especially about the club's ambitions and objectives," Godin added.
"I can't wait to get started and it's great to have a friend in the dressing room. It's the start of a new challenge and a new experience for me.
"I have huge desire to continue competing at the highest level and to continue giving everything I have both on and off the pitch to help this massive club achieve big things."