The Scottish Premiership side were the better team for much of the game as they attempted to overturn a 2-1 deficit from last week’s first leg.
But they fell further behind just before the hour mark when Gerard Gohou netted against the run of play to leave them with so much to do.
Kenny McLean’s goal six minutes from the end game Aberdeen hope of forcing extra-time and they bombarded Kairat’s goal in the closing moments.
Paul Quinn nearly scored the second strike they needed deep in injury time but his header was tipped over and that left the home players distraught at the final whistle.
Nevertheless, McInnes was happy with the effort his team gave and for the way they kept pressing despite often being left frustrated by their opponents.
He said: “I couldn’t ask any more from the players in terms of effort. The pleasing thing for me is that we were clearly a team hell-bent on winning the game.
“The intensity was there and the players deserve a lot of credit for that. There’s a lot of pride and satisfaction.
“I feel as though the team is improving and the fact that every one of the 20,000 in the stadium stayed to a man shows recognition of that.
“I thought privately before the game that the first goal had to come as early as possible because of the way we set the team up with a lot of intensity.
“The first goal was so important, and we got caught running the ball in the wrong area. I thought we were better than them and that we would get through this tie.
“Until somebody says to me at the end of the season, hopefully with a trophy in our hands, that going out of Europe was the best thing that happened to us, it’ll continue to be a disappointment.”