The Bhoys were certainly not outclassed but they never really looked like breaking the deadlock, while the visitors seemed content with a stalemate from the outset.
Gordon Strachan was without Craig Beattie and Maciej Zurawski although Jan Vennegoor of Hesselink passed a late fitness test.
The Italians were also hampered by the absence of key strikers with Ronaldo cup-tied, Filippo Inzaghi injured and Marco Borriello suspended.
Celtic keeper Artur Boruc was forced to make smart save in the opening minutes, diving low to block Kaka's 25-yard drive.
When the hosts hit back Shunsuke Nakamura's long-range free-kick failed to trouble the Milan defence, while Vennegoor of Hesselink was inches away from connecting with a header at the back post.
Kenny Miller then fired a weak effort straight at Milan goalkeeper Zeljko Kalac while at the other end Yoann Gourcuff tested Boruc with a powerful header.
Gilardino then fired just wide before Nakamura's curling effort was saved by Kalac.
Former Rangers midfielder Gennaro Gattuso almost opened the scoring shortly after the restart.
Kaka set him up but Gattuso fired wide when he should have at least hit the target.
Nakamura fired another free-kick tamely wide before Gilardino was booked for diving, claiming he had been tripped by Boruc, when it looked as though he could have beaten the goalkeeper to the ball and had a shot at goal.
Late substitute Thomas Gravesen fired just over the crossbar, but the match eventually ended goalless leaving Milan favourites to qualify for the quarter-finals.