Celtic claimed the scalp of reigning European champions AC Milan thanks to a late goal from Scott McDonald.
McDonald celebrates his late winner for Celtic
Following a dreary hour of play, Celtic captain Stephen McManus forced the ball into the net from a corner kick.
Milan levelled with a Kaka penalty but Celtic snatched victory when Dida failed to hold a Gary Caldwell shot and McDonald netted from close range.
A great win was soured with Dida carried off on a stretcher after a fan ran on to the park to confront him.
And the home team could be in hot water with Uefa, although the Brazilian appeared to suffer little more than a cheeky slap.
Celtic and Milan now sit on three points each, with Shaktar Donetsk leading Group D after a win at Benfica.
A torrential Glasgow downpour made life difficult for both sides and a messy opening period was littered with stray passes and clumsy challenges.
The game's first shot on goal came after 18 minutes and Celtic goalkeeper Artur Boruc gathered a skidding long-range drive from Kaka with ease.
The foul count continued to grow at a bewildering rate and the normally noisy Celtic Park was muted with frustration at the wretched stop-start nature of the match.
Six minutes before the interval, Scott Brown managed to guide a shot on target but Dida was not troubled by the busy midfielder's effort from distance.
The game showed no signs of coming to life until Celtic grabbed a goal from nowhere on the hour.
An ungainly touch from Alessandro Nesta resulted in a corner and Paul Hartley's delivery was bundled into the net by McManus at the back post.
AC Milan were almost gifted an immediate equaliser when Andrea Pirlo's deflected free-kick wrong-footed Boruc but the Polish keeper recovered to awkwardly scoop the ball over his crossbar.
However, the visitors were on level terms soon after, following a foolish foul from Lee Naylor on Massimo Ambrosini.
With the ball sailing harmlessly by, the Italian collapsed when the full-back tugged at him and Kaka sent Boruc the wrong way from the penalty spot.
Aiden McGeady fizzed in a shot that Dida spilled as the game threatened to open up at last.
And the home side were rewarded with a dramatic late winner.
Caldwell fashioned a shot from the edge of the penalty area and the unconvincing Dida pushed the ball straight into the path of McDonald, who gleefully tucked away the easiest of chances.
Celtic: Boruc, Perrier Doumbe (Kennedy 79), Caldwell, McManus, Naylor, McGeady (Nakamura 85), Hartley, Donati, Jarosik (Killen 84), Scott Brown, McDonald. Subs Not Used: Mark Brown, Riordan, Sno, O'Dea.
Booked: Scott Brown, Donati, McDonald.
Goals: McManus 62, McDonald 90.
AC Milan: Dida (Kalac 90), Oddo, Nesta, Bonera, Jankulovski, Ambrosini, Gattuso, Pirlo, Seedorf (Gourcuff 55), Kaka, Inzaghi (Gilardino 76). Subs Not Used: Emerson, Simic, Favalli, Brocchi.
Booked: Ambrosini, Nesta.
Goal: Kaka 68 pen.
Ref: Markus Merk (Germany).