Celtic’s dream of reaching the last eight of the Champions League was crushed by one piece of Brazilian magic last night as Kaká pierced the resistance of Gordon Strachan’s team to decide a taut occasion in the San Siro Stadium with an extra-time winner.

The Scottish champions, cruelly denied a penalty near the end of normal time of the second leg of the leg of the first knockout round when Shunsuke Nakamura was pushed in the back by Massimo Ambrosini, were eliminated by Kaká’s glorious run and clinical finish.

The contest entered extra time when the second leg in the San Siro mirrored the goalless first match in Glasgow a fortnight ago and Milan wasted no time in breaking the deadlock. Just three minutes into the first period, Kaká took advantage of an error by Evander Sno and raced towards the Celtic box where he clipped a left-foot finish beyond Celtic’s goalkeeper, Artur Boruc, who had produced a series of impressive saves but could not keep this one out.

For the 10,000 Celtic supporters who backed their team raucously in the San Siro, there could only be pride that that Strachan’s men had come close to matching the six-time winners almost every step of the way.

Despite only being allocated 4,800 tickets from Milan, thousands more Celtic supporters had found their way into the San Siro Stadium. The request from their club to remain at home had been ignored but the hardline rules — even the press were required to produce their passports — were softened by the fact that fans arriving from Scotland were able to purchase match tickets at a local bank during the day.

Strachan chose to leave out Kenny Miller in favour of the height of Jiri Jarosik, who partnered Jan Vennegoor of Hesselink in a matching 6ft 3in partnership that would clearly need a supply from Nakamura.

The contest swiftly showed it would be far more open than the goalless draw in Glasgow and Celtic required Artur Boruc to make a vital save after just three minutes after Clarence Seedorf supplied Kaká and the Brazil player curled a right-foot shot which was denied by the Celtic goalkeeper who sprawled to his left to touch it wide.

Celtic responded with a move that led to a penalty appeal being wrong rejected. Vennegoor of Hesselink was fouled on the left flank by Daniele Bonera which allowed Nakamura to deliver a free kick to the back post that was met Jarosik. He controlled the ball and thumped it towards goal where it was blocked by the hands of Paolo Maldini but Konrad Plautz, the Austrian referee, refused to listen to Celtic’s claim.

Darren O’Dea is at the other end of the scale to Maldini but the 20-year-old Irishman, in only his third European appearance, showed composure beyond his years as he slid in to cut out Marek Jankulovski’s low cross. Andrea Pirlo then curled a free kick wide of the post.

Celtic’s desire to get a goal was almost their undoing as Milan twice hit them on the counter-attack from their own box. The first, in the 22nd minute, saw Seedorf roll a pass across the face of goal for Filipo Inzaghi but before the Milan striker could sweep it into the net, Lee Naylor made a remarkable interception to steal the ball off his toes.

Then, Nakamura was robbed by Jankulovski on the edge of Milan’s box, Kaká then skipped away beyond Stephen McManus before rolling an angled shot wide.

Yet, Celtic were probing ceaselessly with Aiden McGeady drifting inside and getting on the ball to offer greater creativity. Had the young Glaswegian been more vigilant he could have profited from Vennegoor of Hesselink turning Maldini on the touchline but Oddo snuffed out the threat.

The attacks were swapped with rapidity, with Kaká being denied by O’Dea guarding the line in the 37th minute before Celtic came close after Vennegoor of Hesselink’s diving header hit off Oddo and almost went into his own net. Then Vennegoor of Hesselink just failed to add a touch to Jarosik’s raking cross.

Milan opened the second period with zest and Kaká earned another free kick when tumbled but Pirlo’s effort, though it cleared the wall, was gathered by the arms of Boruc. Then Inzaghi hooked his effort into the sidenet from Jankulovski’s cross.

Perhaps the worst scare that Celtic had came in the 56th minute when O’Dea gave the ball away and Seedorf quickly fed Inzaghi who advanced on goal. Naylor tried to get back and caught the striker right on the edge of the area. The referee ruled that the offence had occurred fractionally outside but the short free kick to Pirlo was charged down.

Celtic sensed that they had to change the tide and fine skill by Jarosik released McGeady, whose pace took him beyond Oddo. The midfield player tried to pick out the in-rushing Vennegoor of Hesselink but Maldini’s frantic slide denied the Celtic striker.

Strachan then introduced Thomas Gravesen for Jarosik just after the hour and it took the Dane just a minute to make his presence noticed. He deftly flicked the ball over Oddo inside the box after a layoff from Nakamura but then saw Maldini block his netbound shot.

Celtic survived a frantic scrutiny in the 70th minute when they headed the ball off the line twice. First, it was Inzaghi who climbed to glance the ball beyond Boruc but McManus cleared away for a corner which was met by Massimo Ambrosini and his header was nodded away by Naylor at the back post. That signalled a taut finale.

AC Milan (4-4-1-1): Dida — M Oddo (sub: D Simic, 116min), D Bonera, P Maldini, M Jankulovski — G Gattuso (sub: C Brocchi, 79), A Pirlo, M Ambrosini, C Seedorf — Kaká — F Inzaghi (sub: A Gilardino, 73). Substitutes not used: Z Kalac, R Oliveira, G Favalli, Y Gourcuff. Booked: Ambrosini.

Celtic (4-4-2): A Boruc — P Telfer, S McManus, D O’Dea, L Naylor — S Nakamura (sub: K Miller, 105), N Lennon, E Sno (sub: C Beattie, 97), A McGeady — J Jarosik (sub: T Gravesen, 62), J Vennegoor of Hesselink. Substitutes not used: M Brown, M Wilson, D Riordan, J Kennedy. Booked: McManus, Naylor, McGeady, Lennon.

Referee: K Plautz (Austria).