Celtic's hopes of Champions League progression are over after they once again wasted chances before losing to RB Leipzig.
Ange Postecoglou's side knew they had to avoid defeat to have a realistic chance of progressing in the competition beyond the World Cup and had early chances through Matt O'Riley, Kyogo Furuhashi and Daizen Maeda.
Giorgos Giakoumakis also went close before Timo Werner headed in the opener for the visitors in the 74th minute.
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Emil Forsberg then fired home late on to secure Leipzig's second win over Celtic in under a week and take them into second in Group F.
While a top-two finish is now beyond Celtic, their hopes of salvaging a Europa League place were boosted by Real Madrid's late equaliser away to Shakhtar Donetsk, who are now four points ahead in third and visit Glasgow in a fortnight.
Celtic rue missed chances once again
Celtic were missing the injured Jota and lost another winger, Liel Abada, to injury three minutes before half-time.
The theme of their European campaign continued inside 60 seconds when Maeda spurned a glorious opportunity as he stooped to head Abada's cross wide from four yards.
Leipzig soon took the sting out of the game, though, with long spells of possession and Dominik Szoboszlai returned David Raum's deep cross beyond the opposite post from a good opening.
Celtic needed good blocks from O'Riley - who was again playing in a deep midfield role - and Moritz Jenz after being troubled on the break.
The hosts came to life after the midway point of the half. Greg Taylor and Sead Haksabanovic caused some panic with balls in from the left and Jenz then headed over the latter's corner.
Leipzig had a major let-off when O'Riley hit the post from 25 yards following a short corner and Taylor scuffed the rebound off the crossbar before Maeda headed wide.
Furuhashi then missed an excellent chance when he sent a free header over the crossbar from 12 yards following a pinpoint cross from Haksabanovic.
Abada went off for James Forrest after sustaining what looked like a knee injury following a fair tackle from Raum.
Leipzig finished the half well. Raum hit the bar with a cross, Joe Hart denied Christopher Nkunku with his foot and Willi Orban sent a free header over from the resulting corner.
Hart pulled off a better stop from Orban early in the second half and Furuhashi failed to make the most of good set-up play from Forrest at the other end.
Celtic were living dangerously. Jenz and Cameron Carter-Vickers put in some important interventions and O'Riley tracked back to make an excellent sliding tackle - and they also got some fortune when Szoboszlai slipped in the penalty box after Reo Hatate lost the ball to leave his defence outnumbered.
Leipzig were enjoying the better of the second-half chances. Right-back Mohamed Simakan and Nkunku missed the target from decent chances before Hart saved from the latter.
Celtic made a triple change midway through the half with David Turnbull, Aaron Mooy and Giakoumakis coming on and they soon had a chance when the Australian cut the ball back for Maeda.
The Japan attacker was having a frustrating night with the ball though and steered a first-time effort wide.
The breakthrough came at the other end after Leipzig worked the ball down their left. Andre Silva got to the byline and cut it back for Werner to head home.
Celtic threatened an equaliser but O'Riley saw a strike palmed away and Giakoumakis headed Forrest's cross wide from close range.
They were made to pay for the latest miss when Werner this time got down the left channel to cut back for Forsberg to find the top corner.
What the manager said
Celtic boss Ange Postecoglou:
"I just want the players to get the rewards for their endeavours.
"I'm trying to get us to play a certain way for a reason but if you don't take those opportunities at this level it's going to hurt you one way or another.
"It's been a story of our campaign so far, it's been a tale of missed opportunities and from my perspective it's just disappointing for our fans and for the players who don't get a reward for their efforts.
"It's about experience and composure at this level and the fine lines and stresses that are involved in playing at the highest level of club football.
"Only experience can hopefully get your players more comfortable playing in those situations."
What's next?
Celtic host Hibernian in the Scottish Premiership on Saturday in a 3pm kick-off.
Three days later, Ange Postecoglou's side go to Motherwell on October 19 in a Scottish League Cup quarter-final.
Then on Saturday October 22, Celtic host Hearts in the Scottish Premiership, live on Sky Sports Football from midday; kick-off 12.30pm.