The Dons fought past Macedonian outfit FK Shkendija and HNK Rijeka of Croatia in their first two qualifying rounds.
But boss Derek McInnes saw his hopes of a place in the group stages dashed when the Kazakh club claimed a 3-2 aggregate win on Thursday night.
Having lost the first leg 2-1, on-form Kenny McLean gave Aberdeen hope when he drew them level at 1-1 in the second leg at Pittodrie with six minutes to go but it was not to be enough.
The defeat means all three of Scotland’s Europa League entrants have now crashed out following the earlier exits of St Johnstone and Inverness.
But McGinn said his side could have given nothing more.
The Northern Irishman told RedTV: “Sitting in the changing room after the game you could see how down all the boys were.
“I thought when we got the equaliser we might have gone on and got a winner to force extra-time. We kept working hard but we probably should have worked their keeper a bit more once we got into the final third.
“I thought the effort was there but we just couldn’t get that second goal we needed.
“Big credit to the fans for turning out in their numbers and selling out the stadium. It was a great atmosphere and it is good to be part of these special nights.
“On another occasion, with a little luck we could maybe have kept a clean sheet. If we had kept a clean sheet maybe we could have got the job done.
“However, we have definitely enjoyed this European adventure and the boys can be proud of themselves.
“We just have to dust ourselves down now and get back to league action.”