Celtic's Champions League campaign has proved a damaging, polar opposite to that of a year earlier. Whereas progress to the last 16 in 2012-13 was earned by a clinical touch in front of goal and obdurate defending, they have failed this time because of wastefulness and generosity.

Only by winning at Camp Nou in the final Group H match can Celtic avoid their lowest ever Champions League points haul. Neil Lennon's team have conceded eight goals in five games while scoring only two. Neither of those came from a striker. Most importantly, they are already consigned to finish in last place.

"Tonight was symptomatic of our entire campaign," Lennon said. "We missed good chances and that has been apparent over the course of our campaign. When chances arrived, we snatched at them. That was the difference. To continue to play in the Champions League we have to improve with the squad we have now, and with recruitment for next year."

This was not in the script for Lennon, who started the evening with aspirations of the knock-out phase for the second year in a row. With half an hour to play, Celtic already trailed Milan by three goals.

As Celtic's inquest opened, Milan's first win in eight outings will ease the pressure on their coach, Massimiliano Allegri. His opposite number at Parkhead has just cause to wonder if his team have gone backwards over the past 12 months.

The reality for them from now until May is the monotony of Scottish domestic football and retaining the title in a league without a legitimate challenger. Milan, who were inspired by Kaká, can glance towards loftier targets.

"This was a very important victory when put in context," said Allegri. "The team has only won once here before, in 1969, so it is a great achievement by these players."

The clear, extenuating circumstance for Celtic's record is the summer sale of key players – namely Victor Wanyama and Gary Hooper – without adequate replacement. The stupidity of his captain Scott Brown, who missed three matches through suspension, hardly helped Lennon's cause either.

Yet in Milan, Celtic last night faced a team in crisis. The Rossoneri have endured a torrid start to their season, a matter not lost on protesting fans after Saturday's home draw with Genoa.

Earlier, a contingent of Celtic's support had produced a selection of banners depicting William Wallace and Bobby Sands asking, among other things, about "the terrorist or the dreamer". Uefa may well take a dim view of such expression, should it deem it political. Those who offered it presumably failed in a day-release request from their time warp.

The opening to this encounter was brilliantly breathless. Derk Boerrigter, handed his first start for Celtic in a Champions League game, should have put his team in front inside two minutes. Instead, the Dutch winger completely miscued his attempt from a Charlie Mulgrew corner.

Then when a Mulgrew shot fell at the feet of an isolated Beram Kayal he delayed, and eventually shanked, his goal attempt.

That, coupled with dire Celtic marking, was soon costly. Kaká was left completely unmarked from Valter Birsa's deep corner and the Brazilian nodded past the stranded Fraser Forster.

Following another spell of Celtic pressure, the visitors could have doubled their lead. Kaká fed Mario Balotelli, who shot only narrowly over. Mulgrew was the next Celtic player to try his luck, nine minutes before the break, but shot tamely and into the hands of Christian Abbiati.

Just two minutes after the restart, Celtic's spurned another excellent chance. Virgil van Dijk was the culprit, failing to beat Abbiati from point-blank range.

Again Milan responded with a goal, Christian Zapata tapping home after a corner was played right to his feet, at the back post, by the substitute Antonio Nocerino.

The lively Balotelli added some Milan gloss, and in what had become typically straightforward fashion. The former Manchester City striker ran on to a long ball from Riccardo Montolivo, shrugged off the attentions of Efe Ambrose, and slotted past Forster.