After losing 2-1 in Kazakhstan last week, Derek McInnes’ team fell further behind in the second half but Kenny McLean’s late header gave them hope of taking the tie into an additional 30 minutes.
Paul Quinn’s header deep in injury time was tipped over to deny them, though, and their exit means Celtic are now the only Scottish club left in European competition this season.
Home manager McInnes didn’t just want a single-goal win before kick-off, speaking instead of his wish to find the net more than once in order to kill off their opponents.
As it was, Aberdeen nearly gave themselves a mountain to climb almost immediately as Kairat went close to scoring twice in the first three minutes.
Their first opening came just 15 seconds in as Danny Ward saved from Gerard Gohou and the on-loan Liverpool keeper reacted again soon after to keep out Stanislav Lunin, who finished poorly.
McInnes’ men quickly settled and after Niall McGinn’s shot was kept out, they felt they should have had a penalty when Peter Pawlett tumbled in the box.
He fell after a shoulder challenge by Yermek Kuantayev but referee Miroslav Zelinka correctly waved away the claim and it remained goalless despite a frantic start.
Aberdeen were having plenty of the ball but Jonny Hayes couldn’t find a way past away keeper Vladimir Plotnikov with three separate attempts, the last of which went closest.
The Scots were stretching their opponents though and Zarko Markovic, Anatoliy Tymoshchuk and Mikhail Bakaev all collected first-half bookings as Kairat took a physical approach.
While that irked most in a crowd of over 20,000, the Kazakhs showed poise at times too and Bakaev’s shot was sneaking shortly before the break but Ward got his angles right to stop it.
He couldn’t keep out the goal that put Kairat ahead after the restart which put them in command of the tie as Gohou struck after Bakaev’s first attempt was kept out.
It took both Aberdeen and their fans time to respond but with more than half an hour to play, there was scope to hit back.
McInnes threw on striker Adam Rooney as he looked for a leveller but Kairat started playing for time and it was hard for the home side to find a rhythm as play was continually interrupted.
Indeed, substitute Serginho almost doubled the lead but hit the crossbar after being set up well by Ukrainian international Tymoshchuk.
That was a turning point as seconds later, with six minutes to go, McLean scored for the third game running as he converted a low, stooping header at the back post to throw his side a lifeline.
Aberdeen threw everything they could at Kairat in the closing stages but five minutes into injury time, Quinn’s effort was touched over and with that their chances of a recovery went too.