This is the third failure for the "Rosoneri" in the last four games, so they are now two points behind Napoli and nine behind leaders Inter, who still have to play their matches.
The "Zebras" took the lead in the 27th minute with an own goal by Davide Bartezagi.
Ten minutes later, Jürgen Ekelenkamp doubled the visitors' lead with an assist from Nikola Zaniolo.
Artur Ata put an end to the match in the 71st minute.
Milan will be away to Verona in the next round, while Udinese will host Parma.