Inzaghi broke the deadlock 21 minutes from the end after knocking in Cafu’s cross from close range to break the record held by German great Gerd Muller as the top scorer in UEFA competitions.

And although Celtic failed in their bid to get the point they needed - Shakhtar’s defeat enabled the Hoops to finish second behind defending champions Milan in Group D.

Gordon Strachan sprung a surprise by dropping Jan Vennegoor of Hesselink from his starting line-up.

AC Milan coach Carlo Ancelotti rested several of his regular players, including Brazilian goalkeeper Dida, with Zeljko Kalac taking his place in goal.

Dida had controversially faked injury at the end of the first game between the sides in Glasgow and was subsequently banned by UEFA – and Ancelotti’s decision tonight kept him out of the spotlight.

Celtic looked nervous and jittery in the opening stages and made unforced errors but Milan failed to punish them.

Artur Boruc did well to anticipate Inzaghi who was ready to fire Milan’s first goal of the evening early on.

In the 15th minute, Stephen McManus gave away a dangerous free-kick after bringing down Inzaghi just outside the area but Milan wasted a good chance with Clarence Seedorf’s effort going straight into the wall.

Celtic were happy to sit back and hit Milan in the counter-attack but lacked the incisive cutting edge to break the Rossoneri defence.

In the 25th minute, Kaka fired wide of the near post and soon after Giuseppe Favalli served Inzaghi inside the area – but his first touch went straight into Boruc’s hands.

Ancelotti was forced to make his first substitution on the half-hour mark with Kakha Kaladze replacing the injured Dario Simic.

Jiri Jarosik was the first to test Kalac with his right-footed effort from a central position proving an easy catch for the keeper.

Milan almost broke the deadlock shortly before half-time with Favalli’s ball finding Seedorf inside the area – but the Dutchman’s powerful strike went high over the crossbar.

Milan picked up where they left off after the restart and Pirlo’s curled free-kick fizzed wide of the far post.

Seconds later, Inzaghi’s angled effort from inside the box went wide as Milan got closer.

At the other end, Scott McDonald failed to find the finishing touch from Scott Brown’s cross from the right.

In the 55th minute Brown missed a golden chance to put Celtic in front when he shot wide after collecting McManus’ long ball.

Milan returned the favour shortly after, with Favalli’s cross from the left finding an unmarked Inzaghi – but his header went straight to the keeper’s hands.

Just past the hour mark, Kaka’s left-footed strike went over, and soon after Strachan made his first substitution, with Vennegoor of Hesselink replacing McDonald.

But it failed to spark Celtic as Milan took the lead shortly after.

Kaka’s pass served Cafu inside the area, and he squared for Inzaghi to fire past Boruc.

Stunned, Celtic attempted to find the equaliser – but it was Milan who had the upper hand.

Kaka’s 25-yard strike forced Boruc to fully stretch to clear.

Strachan made his second substitution with 13 minutes remaining with Maciej Zurawski taking Jarosik’s place.

Milan should have doubled their lead 13 minutes from time, but Boruc made spectacular saves to deny Christian Brocchi and Inzaghi.

Shortly after, Brown warmed the hands of Kalac with a stinging drive which the keeper palmed away for a corner.

Celtic should have equalised with two minutes remaining but Vennegoor of Hesselink’s header from Brown’s cross from the left was off target.

At the death, substitute Evander Sno had to make a goalline clearance from Kaka’s strike – and soon after the final whistle meant both sets of players and fans were celebrating qualification.