The Cameroon international joins in a Bosman transfer, signing a five-year contract.
Onana said: “I am very happy to be part of this great club, of this family that is a great institution. Being here for me is the top and we hope in the future to achieve important goals.
“I spoke with (Samuel) Eto'o several times, he always told me that Inter is perfect for me. That's why when I heard about the offer I never had any doubts.
“Inter have always had good goalkeepers, including the one currently playing. I am calm, it is a great challenge and time will tell if I will be able to win it. But I am very confident."
Onana also sent a message to the Inter fans.
“I know the season has not been what we expected, especially the last game where we failed to become champions, but the next ones will be better.
“We will fight to win everything, here and in Europe, because this team deserves all the best and with the players we have we have to think big. So have confidence in us."