Raphael Onyedika netted a 94th minute winner for Midtjylland as the tie entered extra-time after being locked at 2-2 on aggregate.
Callum McGregor had put the Scottish runners-up ahead early in the second half, but the Danes leveled from Australian winger Awer Mabil on the hour mark.
New Celtic boss Ange Postecoglou told the BBC: “I don’t know what your version of catastrophic is but it certainly doesn’t fit my definition of what happened tonight.
“This is obviously a disappointing night, disappointing we didn’t get through, I can’t fault the effort of the players, I think they worked their socks off, we controlled the game.
“Catastrophic to me means the end. This is far from the end. You are suggesting this is a club falling apart and our season is finished, I don’t see it that way.
“Yes, it’s absolutely disappointing, Champions League football is a priority but I can’t ask any more of the players.”
Elsewhere, Sparta Prague turned around a 2-1 deficit with a 2-0 home win over Rapid Vienna to progress 3-2 on aggregate.
PSV Eindhoven rounded out their comprehensive 7-1 aggregate triumph with a 2-1 win over Turkish giants Galatasaray.
Red Star Belgrade netted five goals in a 5-0 rout of Kairat to turn around their tie and win 6-2 on aggregate.
Olympiacos, Young Boys, Ludogorets, Sheriff Tiraspol and CFR Cluj also progressed on Wednesday.
In the next qualifying round, PSV will meet Midtjylland, Monaco faces Sparta Prague and Rangers go to Malmo FF.