AEK beat 1967 champions Celtic 2-1 in a fervent atmosphere in Athens for a 3-2 aggregate win over the Scottish side while PAOK held out for a 0-0 draw at Spartak Moscow after edging the Russian outfit 3-2 in the first leg.
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.
Sinclair pulled one back in the 78th minute, inspiring hope among Celtic fans, but it proved to be merely a consolation goal as the Bhoys ran out of time.
Dropping down to the Europa League, where they will play Latvian side Spartaks Jurmala or Lithuanian side Suduva, is a huge blow for Celtic who will miss out on the prestige - not to mention the vast sums of money - to be earned in the Champions League. AEK will face MOL Vidi in the final round of Champions League qualification.
Ajax boss Erik ten Hag was not content with its two away goal lead, and crushed its Belgian visitors with goals from Klaas-Jan Hunterlaar, Matthijs de Ligt, and David Neres. "It's important that we have European football this year".
Fulham had a bid rejected for the 27-year-old last week.
"My concentration really until I get back is only with the players that are here".
"I've always said we needed to strengthen (in the transfer window) and hopefully we can do that now".
"But he wasn't here and we look at the performance without him and I can't think any more than that".