The defender joined Milan from VfL Wolfsburg in the final minutes of the summer market.
The 19 year-old beamed: "Yesterday everything happened so quickly, it was incredible. I still can't quite grasp what's happened in the last 24 hours.
"So much has happened, it's an incredible feeling. I'm very grateful to be here and I'm happy to be part of this club."
Odogu moves to Milan for €10m and has signed a deal to 2029 with an option for another 12 months.