When it mattered most, Celtic produced their worst performance in nine European outings this season against Benfica and undid their glorious accomplishment against Barcelona when they conceded a needless opener to Ola John early in the match and failed to defend a corner midway through the second half.
The decisive moment came when Nemannja Matic's delivery was headed on by Luisao to Ezequiel Garay, for a volleyed finish that means that Neil Lennon's side must now beat Spartak Moscow at home in the final round of matches and hope that Barca take points from Benfica.
Not even Georgios Samaras's seventh goal of the season - six of which have been scored in European games - could inspire his team-mates, who looked leaden and dull witted. They will certainly play in Europe after Christmas, but for Celtic the Europa League would be a let-down after having come so close to the last 16.
The travelling support was boosted when Scott Brown was declared fit to play on the right of a midfield quartet that included Victor Wanyama, Joe Ledley and Charlie Mulgrew. Gary Hooper, too, was included after injury. Benfica's eagle - which had deserted the stadium earlier this year - was back, too, and it set the example for the swoop that gave Jorge Jesus and his players the start they wanted.
Celtic had looked off the pace from kick-off, as Samaras demonstrated when he was caught in possession by Oscar Cardozo, who slashed his shot over the top. That warning went unheeded and in the seventh minute Celtic paid the price when Adam Matthews was unable to intervene as Benfica surged down the left to find Cardozo inside the box.
His attempt at a cutback came off Brown and fell to Ola John a dozen yards out. The striker's shot was aided the inside of Mikael Lustig's right leg, from which it spun behind Fraser Forster and into the goal.
If the build-up had resembled a pinball machine, there was no question about which side had been in charge and even when Celtic sprung a counter-attack they were sabotaged by a poor offside call against Hooper.
But a gift from the gods arrived for Celtic, whose first corner arrived in the 32nd minute. Mulgrew stepped up to take it. Astoundingly, they allowed Samaras to loiter, entirely unattended at the back post. Mulgrew promptly deposited the ball straight on the Greek's forehead and from a yard out the forward bounced his effort into the roof of the net.
Any thoughts that Celtic would employ the interval to calm their nerves was banished seven minutes after the restart when Matthews had to block on the line from Andre Almeida, who had fulfilled Lennon's pre-match prediction that the Benfica full-backs would push forward.
The crucial moment arrived with 18 minutes left to play, when Celtic conceded another corner that indicated Benfica's superior possession. This time their concerted efforts could not deny Garay who was unmarked when he latched on to Luisao's knock-down header.
The centre-back's strike proved to be the winner, but the damage could have been more severe but for interventions by Forster and the crossbar, when it deflected a blistering drive form Silvio. Incredibly, Celtic could have stolen their precious equaliser in injury time when a Kris Commons cross found Tony Watt but the teenager's raking effort beat Artur but also swept beyond the far post.
Benfica (4-3-3): Artur; Andre-Almeida, Luisao, Garay, Melgarejo; John, Matic (Maxi Pereaira 78), Lima (Gaitan 75); Perez, Oscar Cardoxo, Salvio (Jardel 90). Subs: Lopes (g), Cesar, Nolito. Booked: Melgarejo.
Celtic (4-4-2): Forster; Lustig, Wilson, Ambrose, Matthews; Brown (Commons 60), Ledley (Watt 80), Wanyama, Mulgrew (Kayal h-t); Hooper, Samaras. Zaluska (g), Miku, Lassad, McCourt. Booked: Samaras, Ledley, Wanyama.
Referee: V Kassai (Hungary)