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.
Celtic will now head into the Europa League playoff round and face either Latvian or Lithuanian opposition.
Rodgers bemoaned some slack defending for their Champions League exit.
"I thought a lot of our build-up play in that final third was worthy of something from the game, but you can't defend that softly and in the end we were punished for it".
"I've always said that we needed to strengthen, no matter how good we've been over the last couple of years".
"In the main, there weren't too many opportunities against us but you can't concede that easily and unfortunately it's cost us".
BRENDAN Rodgers has insisted Dedryck Boyata is fit and was not taken to Athens in case it hurt the spirit within the squad.
Boyata's agent Jacques Lichtenstein claimed the Parkhead club had broken promises to his client, who was the subject of a failed £9million bid from Fulham before the English transfer window closed.
The ongoing and very public row between the Celtic defender Dedryck Boyata and his club has escalated yet further today, with both sides digging in at their positions over the Belgian defender's supposed refusal to play for the club following dispute over an aborted move to Fulham.
Asked if he felt frustrated on a night where defensive inadequacies proved costly, he said: "Yes, of course".
"My concentration really until I get back is only with the players that are here".