Derek McInnes’ side appeared to be in real trouble in the tie after just 22 minutes thanks to first-half goals from Mikhail Bakaev and captain Bauyrzhan Islamkhan.
However, substitute Kenny McLean’s 69th minute strike will have given the Dons confidence heading into the return fixture at Pittodrie next Thursday.
The hosts enjoyed the better of the early possession and Aberdeen looked understandably a little off the pace having made a 3,400-mile journey to Almaty.
Kairat took the lead after 13 minutes when Bakaev fired past Danny Ward in the Aberdeen goal following some excellent build-up play.
Derek McInnes’ side started to come into the game and had chances through Jonny Hayes and Niall McGinn but fell further behind nine minutes later thanks to an outrageous 30-yard strike from Islamkhan which sailed past Ward into the top corner.
Aberdeen forced corners and enjoyed possession but did not share their opponents’ cutting edge in front of goal in the first half.
Kairat’s Brazilian attacker Isael could have scored either side of the interval but was brilliantly denied twice by Ward while Djorde Despotovic fired just wide of the goalkeeper’s right-hand post from a tight angle on the hour mark.
However, Kairat will now face an entirely different proposition when they travel to the Granite City next week thanks to substitute McLean.
The former St Mirren player bagged his first Aberdeen goal in the previous round against Rijeka and he was on target again here when he got on the end of McGinn’s blind cutback across the box to smash the ball into the net.
There was a real scare for Aberdeen in the 90th minute, however, when Stanislav Lunin’s bouncing effort from the edge of the box crashed off the bar.
McInnes told the club website: “I did not think we were brave enough when we did not have the ball, we did not do enough to impose ourselves on the game. The response from my team was exactly what I thought it would be. We did not roll over. We came out fighting.
“If we play for the full 90 minutes in the second leg the way we played the last hour of the game here then I feel we have an excellent chance to go through. But we still have it all to do.
“I think Kairat are still the favourites as we still have to score. But if we do score then one goal could be enough. We will have 20,000 supporters ready for the game.”