The Scottish champions looked to be cruising into the league stage when Callum McGregor put them 4-0 up on aggregate against LASK with a 21st-minute deflected strike on Monday (Tuesday AEST).
However, they went on to endure one of their all-time European lows with a 5-1 defeat in Linz delivering a £20million-plus ($A30 million-plus) blow.
The Hoops had other chances in the first half but the Austrian champions gave themselves a lifeline when Moses Usor pounced on a loose pass from Benjamin Nygren to level on the night in the 32nd minute.
The tie began to spiral out of Celtic's control early in the second half as Robert Ljubicic and Samuel Adeniran netted from outside the box.
Martin O'Neill's side looked like they would cling on but Florian Flecker nodded home from close range in added-time to spark an additional 30 minutes.
Celtic had disappeared as an attacking force at halftime and Adeniran delivered the knockout blow that sends the Hoops into the Europa League.
They exited on penalties at the play-off stage last season after drawing a blank against Kairat Almaty. Fans will this year question the decision to sell attacker Arne Engels just ahead of the first leg in Glasgow.
The competition's feel-good story Bodo/Glimt are back for a second season.
The Norwegians eased past Dutch newcomers NEC Nijmegen, winning 3-0 at the Arctic Circle for a 6-1 aggregate score.
Sabah of Azerbaijan - just nine years after being formed - were just seconds away from elimination against Israeli champions Hapoel Beer-Sheva, who won the first leg 2-1, when substitute Tellur Mutallimov scored in the last minute of added time to force extra time.
Sabah's Gabon striker Orphe Mbina then headed home in the 101st minute and Joy-Lance Mickels added another in the 115th to make it 5-2 in the second leg and 6-4 on aggregate score.
With the AP.
11°C