In their first match in the Uefa Youth League, Marco Pinato opened the scoring for the hosts with a free-kick six minutes into the match, but Jamie Lindsay headed home a well-deserved equaliser just before half-time.
A soft penalty decision gifted the home side the lead not long after the restart, however, before Krisztian Tamas put the game beyond reach on 75 minutes.
Stevie Frail and John Kennedy named a mix of youth and experience in the squad, with established central defender Stuart Findlay wearing the captain's armband.
The young Hoops began the match brightly, dominating the opening spell, but it was AC Milan who took the lead early on.
Pinato took advantage of a soft free-kick decision awarded against Findlay 25-yards out, and curled it home.
Celtic responded well, though, when Paul McMullan sent a short corner through to Liam Henderson. A cheeky one-two, with the latter back-heeling it, saw McMullan crossing the ball to the back post.
Lindsay charged in to make contact and did well to head home the well-deserved equaliser.
But another soft decision against Findlay gave AC Milan another goal five minutes into the second half. The defender was deemed to have fouled substitute, Michael Fabbro, Pinato calmly converting from the spot.
Despite Celtic's composed performance, the Italians dealt a killer blow on 75 minutes when Maks Barisic rolled the ball into the path of Tamas with a back-heel and the left-back took his time to smash home.