Following on from the clubs 1-1 draw away to CFR Cluj last week, Neil Lennons Bhoys suffered the worst possible start as Ciprian Deac broke the deadlock in Glasgow.
James Forrest and Odsonne Edouard were able to hit back and put Celtic back in the driving seat, but Billel Omrani scored either side of Ryan Christies effort to give Cluj the advantage, while substitute George Țucudean rounded off the scoring in added time to dump the Scottish champions out of the Champions League.
It was a similar story for other big clubs across the continent as FC Porto, İstanbul Başakşehir and FC Basel also saw their journey in Europes elite competition end during the qualifying rounds.
Portuguese champions Porto threw away their one-goal lead to Krasnodar and conceded three goals at the Estádio do Dragão, eventually being knocked out by the Russian club on away goals.
İstanbul Başakşehir couldnt convert any of their 11 shots on goal in Greece as Olympiacos scored twice to register a 3-0 aggregate win, while Basel collapsed on the road to Austrian side LASK.
Elsewhere in the third qualifying round, Apoel Nicosia, Rosenborg, Club Brugge and Dinamo Zagreb joined the list of teams who played the second-leg away from home to qualify.
Red Star Belgrade - also known as Crvena Zvezda - took Copenhagen to a penalty shoot-out, which saw a stunning nine spot-kicks spurned before the Serbian giants eventually won the match, with Denmark Under-21 international Jonas Wind missing the decisive effort.