Neil Lennon's men spurned more than their fair share of chances and were eventually punished when the hosts rallied in the final minutes of their Champions League opener.
Cristian Zapata's strike deflected off the Bhoys defender to put the hosts ahead after 82 minutes and, with the visitors pushing for a way back into the game, the Ghanaian doubled the Italian side's lead three minutes later to get the Rossoneri off to a winning start in Group H.
The well-documented Milan injury list meant that Massimiliano Allegri was forced into a number of changes, with Stephan El Shaarawy, Kaka, Giampaolo Pazzini, Riccardo Montolivo and Ignazio Abate among the absentees. Slovenia international Valter Birsa made his debut for the club while Antonio Nocerino partnered with Nigel de Jong in midfield for the clash.
Celtic had injuries of its own to contend with and was unable to name James Forrest or Joe Ledley in its match-day squad, although Adam Matthews shook off an ankle problem to start.
Mario Balotelli was quickly among the action, volleying an effort at Fraser Forster after three minutes, but the Celtic goalkeeper was equal to the task as he palmed the strike away.
Celtic's first chance came from a indirect free kick inside the area following a back pass, but Charlie Mulgrew saw his shot deflected over the crossbar. Alessandro Matri's header then forced another good save from Forster before Nocerino fired narrowly over just after the half-hour mark.
Giorgos Samaras had the first opportunity of the second half for Celtic, with the Greek firing just past Christian Abbiati's post. Muntari then headed just over for the hosts a minute after the hour mark.
After a lengthy period of Milan pressure, Anthony Stokes crashed a free kick against the crossbar for the visitors 10 minutes from time with Abbiati beaten.
Two minutes later, however, Milan had its lead courtesy of a Zapata effort from outside the area that deflected off Izaguirre and in.
Muntari was then on hand three minutes later to make it two, bundling home from close-range after Balotelli's free kick had come back off the post to ensure that Milan came away with the three points.