The forward impressed against the Rossoneri in Milan's 3-2 victory in the third round of Europa League qualifying last week, scoring one and assisting another.
Hauge, who will wear the number 15 shirt, landed in Milan earlier this week and completed a medical on Thursday (AEST).
Speaking prior to his switch this week, the 20-year-old Norwegian told TV 2: "I feel I can manage to claim a place in that Milan side.
"It was a step I felt ready to take. I need to thank Glimt for the cooperation we have had in the last days."
After rising through the youth ranks to make his first-team debut for his hometown club in 2016, Hauge scored 35 goals in 118 appearances for Bodo/Glimt either side of a loan stint at Aalesund.
Milan confirmed the deal a day after Lucas Paqueta was allowed to join Ligue 1 outfit Lyon in a deal reported to be worth €20million.