The Scottish club were on course for a first win in Germany after fighting back with an audacious Josip Juranovic penalty – awarded after a VAR check – and a second-half Jota strike.
But Robert Andrich, whose header had given the hosts an early lead, equalised in the 82nd minute and Moussa Diaby thumped in the winner to secure top spot for Leverkusen.
It leaves Celtic four points behind second-place Real Betis, who beat Ferencvaros 2-0, with the Spaniards visiting Glasgow next month in their final group fixture.
Celtic were already assured of third place and a Europa Conference play-off spot, where they will face a runner-up from that tournament, but the nature of this defeat will sting after a promising performance.
The Scots were taught a harsh lesson in finishing when ruthless Leverkusen racked up a 4-0 win in Glasgow in September, but that proved a turning point for Postecoglou’s men, who had won eight and drawn one since.
They had goalkeeper Joe Hart’s heroics – and the woodwork – to thank for a turnaround which would have made history on the club’s 13th visit to Germany.
But Leverkusen’s barrage of late pressure overwhelmed Celtic, who have conceded the highest number of goals in the group stage as their defensive frailties were again laid bare.
They fell behind amid a flurry of early home pressure, having not heeded an earlier warning when Andrich rose unchallenged to head over.
The midfielder made no mistake second time around, running from the edge of the box to meet another Florian Wirtz corner and guide a header in at the far post.
Celtic staged a spirited response, showing composure in possession and pressing with intensity, but were struggling to make inroads in the final third.
When they fashioned a first effort at goal after 40 minutes, it provided the equaliser. Jota swung a dangerous cross into the penalty box and goalkeeper Lukas Hradecky got nowhere near the ball as his attempt to punch clear caught Kyogo Furuhashi flush in the face.
Juranovic – back on penalty duties with Giorgios Giakoumakis injured – was calmness personified with a chipped spot-kick down the middle that kissed the bar on its way in.
The woodwork smiled on Celtic again to keep them level before half-time. Juranovic overran the ball on the edge of the box and Diaby rattled a ferocious angled drive off the near post before Wirtz stabbed the rebound off the opposite upright.
Another scare followed swiftly after the restart, with Hart blocking Diaby’s effort with his legs then diving at the feet of Amine Adli to prevent the winger tapping in.
It was an incredible double save. And Hart was not done there, sparking the move which gave Celtic the lead with a throw out to James Forrest in the opposition half.
The visitors worked the ball slickly into the box where former Celtic full-back Jeremie Frimpong’s slip allowed Kyogo to lay the ball off for Jota to whip a shot past the helpless Hradecky.
With half an hour to go, Celtic were dropping deeper but keeping Leverkusen out as Hart parried an Adli strike over the bar.
The former England international made another terrific stop to divert Adli’s shot over with a foot, even though the 21-year-old looked yards offside, and Celtic were made to rue that decision in the next phase of play.
Nadiem Amiri cushioned a Wirtz cross back to Andrich for an empathic finish through Hart’s legs for the equaliser.
Leverkusen kept up the momentum Diaby lashed home a bouncing strike on the edge of the box from a delicious scooped pass from Wirtz, before Celtic’s David Turnbull had a close-range effort tipped wide in the final seconds as his side’s exit was confirmed.
Man of the match – Cameron Carter-Vickers
What did we learn?
This was a far more controlled Celtic display than their hammering by Leverkusen in Glasgow, but Postecoglou’s side were undermined by their failure to protect a lead – or cling on a for a point – in the final 10 minutes.
While averaging two goals scored per game in the group, their 13 conceded is far too many and ultimately the reason why they drop down to the Europa Conference.
What did they say?
Celtic manager Ange Postecoglou: “We weren’t able to control the game the way we wanted because they have quality in their side and they’re always a threat on the counter-attack so we had to be a bit wary.
“We hung in there and Joe (Hart) made some good saves. We just couldn’t hang on in the end.”
Celtic host Aberdeen in the Scottish Premiership on Sunday (15:00 GMT) and complete their Europa League group campaign at home to Real Betis on 9 December (20:00).
Bayer Leverkusen have never lost a home Europa League group stage match in 15 previous games (W12 D3) – the most of any side without losing in the competition’s history.
Celtic have scored two or more goals in each of their last four away games in the Europa League, but have gone on to win only one of those matches (L3), beating Ferencvaros 3-2 at the start of the month.
Celtic have conceded at least once in 26 of their 28 away Europa League games, conceding in each of their last 11 such matches in the competition, since a 1-0 win in Rosenborg back in November 2018.
Florian Wirtz has been directly involved in ten goals in 13 Europa League appearances for Bayer Leverkusen (five goals, five assists).
Robert Andrich has scored three goals in his last three appearances for Bayer Leverkusen in all competitions, more than he netted in his first nine appearances for the club this season (two).
Celtic’s Josip Juranovic has had two attempts on goal in this season’s Europa League campaign, both penalties which he converted