Scott Sinclair headed in late on, but it was not enough to save Celtic, whose Champions League adventure came to a premature end in Athens.
Celtic's hopes of reaching the Champions League group stage are over after a 2-1 defeat in Greece on Tuesday saw AEK Athens run out 3-2 aggregate victors, putting Brendan Rodgers' future in doubt.
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 club did not have enough to beat the Greek champions.