Piero Hincapie, the 19-year-old Ecuador defender, gave the visitors the crucial lead in the 25th minutes before 18-year-old star midfielder Florian Wirtz added a second in the 36th minute.
Despite strenuous efforts by Celtic to get back into the match – and they had several chances throughout – Lucas Alario scored from the spot just before the hour mark before substitute Amine Adil hammered in a fourth in added time.
After two defeats in their opening two group games, both Celtic and Ferencvaros are bottom of Group G without a point and they play each other next in a double-header.
Postecoglou said: "That was the disappointing thing tonight, I thought we fell away at the end and there was no need to.
"We were still creating chances and if you get a goal you are back in the game, at least you don't get the score blow out because you want to give yourself a chance in the competition.
"We have to win both these games if we want to stay alive so it is a good challenge for us.
"If you lose 4-0 the overwhelming feeling is that it has been a disappointing night, disappointing for the fans who I thought created an unbelievable atmosphere.
"We have to take that disappointment and use it to as motivation to set things right in the next game.
"In many respects we created some of the issues for ourselves.
"We had some unbelievable opportunities, they are a good side with some quality players and they took their opportunities but we helped them as well by making some mistakes and gave them the advantage.
"At this level you are going to get punished for your mistakes and you have taken your opportunities and we didn't.
"Despite suffering his biggest defeat since taking over as Celtic boss, the former Australia boss insists he will continue with his attacking instincts. He said: "Yes, I don't see any reason why not to.
"They are a quality team and there is always a possibility that you will get punished.
"I think we created enough opportunities to cause our own damage tonight and we didn't.
"It shouldn't have been a 4-0 game
"We fell away at end through lack of discipline and that is unacceptable.
"Bayer Leverkusen boss Gerardo Seoane paid tribute to keeper and captain, Lukas Hradecky, who made a series of saves. He said:
"It was a good job well done. We did what we intended to do, we made some mistakes but it was a sensational game from Lukas. We know he is in great form and it was a great performance form him. You can't underestimate Celtic's individual class. Overall we deserved to win."