The hosts looked nervous amid the pressure of being knocked out of the Scottish League Cup by arch-rivals Rangers last weekend and being dumped out of the Champions League play-offs last month.
The visitors took control of the game after half-time, with two quick-fire goals, and Martin O'Neill was in no mood for excuses after the game.
"It's not the start we wanted. But, we know we have seven games to get some points on the board and our focus now is on Sunday and Rangers again," O'Neill said.
"This defeat was, in many aspects, another European lesson for us. Now we need to be ready for Sunday.
"We've started off fine domestically, but this was a big game for us, and we failed.
"It's on me to show leadership. I've had plenty of low moments, but these two losses hit hard; there's no doubt about that."
Celtic are next in action against Old Firm rivals Rangers in the Scottish Premiership.