The Scottish champions lost 4-1 to Legia Warsaw as Red Bull Salzburg and Zenit were stunned and Lille secured a priceless win against Grasshoppers
The Scottish champions, who won this tournament under its former guise of the European Cup in 1967, got off to a superb start when Callum McGregor put them ahead in the eighth minute.
But that was as good as it got for Ronny Deila's men as their gameplan quickly collapsed.
Winger Miroslav Radovic equalised two minutes later and then added a second in the 36th minute, before Celtic saw Efe Ambrose shown a straight red card a minute before the break.
Ivica Vrdoljak subsequently sent a penalty wide for the hosts just before the hour mark, but Celtic were unable to heed the warning as Michal Zyro did make it three in the 84th minute.
Vrdoljak then saw a second spot-kick saved by Fraser Forster three minutes from the end, only for Jakub Kosecki to add a fourth in stoppage time.
There were mixed fortunes for the other former winners involved at this stage, as 1986 champions Steaua Bucharest claimed a useful 2-2 draw against Aktobe, although they led twice in Kazakhstan.
Feyenoord, meanwhile - champions in 1970 - went down 2-1 at home to Beskitas and now have it all to do in a tricky second leg in Turkey.
Red Bull Salzburg suffered a shock 2-1 defeat in a feisty clash with Qarabag which saw three men sent off.
Danilo Dias put the Azerbaijanis ahead in the second minute, and Salzburg's task was made even harder by defender Christian Schwegler's dismissal 32 minutes later.
The numbers were evened up three minutes before the break, Admir Teli shown his second caution, and Salzburg levelled the scores with 13 minutes to go courtesy of Jonatan Soriano.
Qaraba's Reynaldo was at the centre of the action towards the end, scoring the winner in the 86th minute but then getting himself sent off two minutes later as the hosts held on to take an advantage back to Austria.
Sebastien Corchia and Ryan Mendes earned Lille a 2-0 win at Grasshoppers to put them in the driving seat ahead of the second leg of their tie, while Lukas Gikiewicz's strike means AEL of Cyprus will take a narrow 1-0 lead into their decider against Andre Villas-Boas' Zenit in Russia.
Marcelo Brozovic earned Dinamo Zagreb a 1-0 win away at Aalborg, while a goal deep into stoppage time from Damjan Bohar gave Maribor victory by the same scoreline at home to Maccabi Tel Aviv.
APOEL hold the upper hand over HJK after coming back to draw 2-2 courtesy of two away goals in Finland, while Dnipro were held 0-0 at home by Copenhagen.
Elsewhere, the games between Standard Liege and Panathinaikos and Ludogorets and Partizan Belgrade also finished goalless.