Parkhead substitute Scott Sinclair pulled a goal back in the 78th minute but despite a grand finale the Scottish champions could not fashion a second which would have put them through on away goals.
The Dutch side's reward is an intriguing clash with Dynamo Kiev, who eliminated Slavia Prague 3-1 on aggregate after goals by Benjamin Verbic and Artem Besedin fired the Ukrainian side to a 2-0 home win over the Czechs.
Earlier, Boyata released a video on Instagram showing him receiving treatment, accompanied by a caption saying: "Unfortunately I could not play tomorrow because I'm not yet 100% operational".
"But he wasn't here and we look at the performance without him and I can't think any more than that". "We gave AEK a lot of problems but you have to defend properly and we didn't do that".
Celtic will next face either Latvians Spartaks Jurmala or Lithuanian outfit Suduva for a place in the Europa League group stage.
Celtic took control, with Tom Rogic and Griffiths both having luckless efforts on goal before the latter sent a fine Callum McGregor cross inches past the post from eight yards.
Sutton claimed the club had splashed out on expensive "disco lights" but failed to make the right kind of signings to bolster Brendan Rodgers' squad. "I've always said we needed to strengthen (in the transfer window) and hopefully we can do that now".
"First of all, we have to get through that final play-off game and ensure that we have European football".
After drawing the first leg 1-1 in Glasgow, it was all to play for in the Greek capital tonight, but they did not step up to the plate.