There had been suggestions pre-match that former Liverpool coach Rodgers could be tempted to walk away if Celtic failed to make it through the qualifiers and, although Scott Sinclair made sure of a gripping finish, the Scottish champion, which fielded Australia star Tom Rogic, did not have enough to beat its Greek counterpart.
The first-leg draw meant Celtic already had it all to do in Athens before sloppy defending presented AEK with an early chance that was taken with aplomb, Rodrigo Galo blasting home from the centre of the area after a Niklas Hult cut-back was not intercepted in the sixth minute.
The home side had a bit of a wobble around the half-hour mark, as Celtic began to take charge and look the more threatening, Leigh Griffiths just missing the target from Callum McGregor's low cross 10 minutes before the break.
But Celtic continued to look susceptible to the counter and it was not a huge surprise when AEK scored a second in the 50th minute, as Marko Livaja nodded in from close range after Vassilis Lambropoulos headed a free-kick down to the former Inter Milan man.
The visitors were forced to pile men forward as a result and eventually pulled one back 12 minutes from time through Sinclair's well-placed header, but AEK managed to hang on.
Celtic will now go into the UEFA Europa League, where it will face either Lithuania's Suduva or Spartaks Jurmala of Latvia.
AEK will face MOL Vidi in the final round of UEFA Champions League qualification.