With Ajax beating Barcelona 2-1 at the Amsterdam ArenA, the Scottish champions do not even have the safety net of the Europa League this season.
Brazilian star Kaka put the visitors ahead in the 13th minute when he headed a Valter Birsa corner past Hoops goalkeeper Fraser Forster.
Celtic defender Virgil van Dijk missed a sitter at the start of the second half when he blasted a shot straight at Milan goalkeeper Christian Abbiati, a minute before Rossoneri defender Cristian Zapata made it 2-0 from close range.
Mario Balotelli, in one of his more productive moods, scored a third with a drive on the hour mark to compound the misery of the home side who were made to pay for missing several chances.
Celtic remain bottom of the pool with three points from five games and their final fixture at the Nou Camp next month will be for pride only.
Bhoys boss Neil Lennon was disappointed with the nature of the goals his side conceded.
“Disappointed with goals conceded from corners. That was most the disappointing aspect, and the chances we missed,” he said.
“We’ve competed and competed again tonight, but it’s mainly the defensive part of the pitch we're talking about.
“We all enjoy the Euro experience but we'll learn from this campaign and we will do everything we can to come back next year and compete.”
Defender Mikael Lustig bemoaned Celtic’s sloppiness: “I don’t think we’ve let one goal in from set-pieces this season and now we do it in the most important game of the season.
“It’s not good enough.
“We’ve conceded two or three goals from corners in two years and I don’t think it’s the formation that’s wrong.”
AC Milan arrived in the east end of Glasgow in some turmoil having won just three of their 13 Serie A matches this season and without a win in seven.
Lennon had warned against reading too much into the poor form of the Rossoneri — a warning which proved to be right.
The visitors hinted at danger early on when Balotelli drew a shot wide of the target.
Celtic had an even better chance in the ninth minute when Kayal found himself with the ball at his feet in front of goal following a shot from Mulgrew, but the Israeli midfielder hesitated, allowing Zapata to challenge before he skewed his drive wide.
The Hoops looked capable of getting the win they so badly needed but their fans were stunned into silence at Kaka’s goal. Unmarked at the six-yard box when Birsa’s corner came over from the right, the former Real Madrid player had the easiest of tasks heading past Forster.
Celtic responded as expected but Milan remained potent.
In the 36th minute, after Kris Commons was crowded out at the edge of the Milan penalty area, the ball broke kindly to Mulgrew but his left-footed shot was too weak to bother Abbiati.
Balotelli then had the ball in the net after breaking clear of the Hoops defence but the flag was up early for offside.
The game turned in a minute early in the second period.
In the 47th minute, when Mulgrew’s free-kick was blocked by the wall he lobbed the ball into the box to find the unmarked van Dijk but the Dutchman blasted a volley straight at Abbiati.
A minute later the Hoops were two behind and on their way out of Europe.
Birsa’s lofted corner from the right was met by substitute Antonio Nocerino, on for Ignazio Abate, and he steered the ball across goal to the back post where Zapata was in perfect position to knock it in from a yard out to deliver a hammer blow to the Hoops.
Celtic looked as if they had got themselves back into the game soon afterwards when Van Dijk headed in a Ledley cross but the offside flag was up.
When Balotelli escaped the Hoops defence in the 60th minute to pick up a Montolivo pass and knock number three past Forster from close range it truly was all over for the Scottish champions.
CELTIC: Forster, Lustig, Ambrose, van Dijk, Izaguirre, Boerrigter, Mulgrew, Kayal (Ledley 30), Forrest (Rogic 80), Commons (Stokes 65), Samaras.
AC MILAN: Abbiati, Abate (Nocerino 48), Zapata, Bonera, Emanuelson, Montolivo, De Jong, Poli, Birsa (Constant 72), Kaka (Robinho 80), Balotelli.
Ref: Cuneyt Cakir (Turkey).