The Bhoys were 4-1 down after the first leg and lost 2-0 at Murrayfield on Wednesday evening to ensure that they would be playing in the Europa League this season.
The heavy defeat will throw manager Ronny Deila's future into doubt and the club are poised to lose some of their star players, such as goalkeeper Fraser Forster.
Celtic had earned plaudits and prize money from ventures into - and, once, past - the group stage of the Champions League under Neil Lennon, who left his post at the end of last season but Deila will now have to focus on lower targets.
Michal Zyro put the first goal past a limp home side with 10 minutes left of the first half, picking up on a long ball and holding off his marker to fire past Forster.
Michal Kucharczyk had the last word just after the hour, going around the keeper to score, but Celtic's hopes had long since been extinguished.