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.
Rodrigo Galo scored after six minutes for AEK and Marko Livaja made it 2-0 five minutes after half-time.
"Unfortunately, I could not play because I'm not yet 100% operational", the 27-year-old said on Instagram.
"But he wasn't here and we look at the performance without him and I can't think any more than that".
Brendan Rodgers, who had taken the Hoops to the group stages of the competition in his first two years in charge, had admitted the third qualifying round tie in the Olympic Stadium was his biggest qualifying challenge to date following the 1-1 draw in the first leg at Parkhead last week.More news: NASA to launch car-sized spacecraft Parker Solar Probe
"We gave ourselves a chance and threw everything at it".
"It would've been great to have had him here but he's not", added Rodgers.
Celtic drop into the Europa League play-off round, where they will face Latvia's Spartaks Jurmala or Suduva Marijampole of Lithuania.
"I always say you shouldn't have to lose to learn but when you do lose you have to learn so that is our message and we get back to the training field and try to improve".
Elsewhere, two-time European champions Benfica sealed a 2-1 aggregate victory over Fenerbahce.
And ahead of the return leg in Greece, they were stunned 1-0 in the Scottish Premiership by Hearts, with Kyle Lafferty's goal the difference between the sides.