Gordon Strachan's side held the much-fancied Italians to a goalless draw at Parkhead two weeks ago and repeated the feat for 90 minutes at the San Siro with a performance full of courage and commitment.
But Kaka settled a tight and tense last 16 second leg meeting with the winner three minutes into stoppage time.
The Brazil international showed his class by slipping the ball through the legs of on-rushing goalkeeper Artur Boruc after running through the heart of the Celtic defence.
Earlier in the match, Kaka sliced wide from 15 yards as Celtic failed to clear a corner in the opening minutes.
Celtic left-back Lee Naylor was in the right place at the right time midway through the first half as he cleared the impressive Brazilian's low strike off the line.
AC Milan continued to boss the match and Filippo Inzaghi could have broken the deadlock but drilled his shot into the side-netting early in the second half.
The Italy international was felled in the act of shooting by Naylor but the referee ruled the infringement had taken place right on the edge of the box and waved away Milan's penalty appeals.
Boruc finger-tipped away a dangerous Clarence Seedorf free-kick as the home side pressed for a winner in normal time, before Kaka crashed a shot from 18 yards off the crossbar.
But having battled so hard for 90 minutes, Celtic were ultimately undone by the brilliance of Kaka three minutes into extra time.
Milan's outstanding performer slipped away from Neil Lennon on the halfway line and powered past the challenge of Stephen McManus before coolly sliding home his shot.
Celtic threw on strikers Craig Beattie and Kenny Miller late on but failed to find a way through a well-drilled home defence as Milan played out time comfortably to secure their place in the last eight.