The Finnish champions were in a strong position after the first leg a week ago, notching two second-half goals to leave Kuopio with a 2-0 lead.
But that lead was soon in jeopardy in Almaty, as nine minutes in, the hosts put themselves ahead on the nigh. A corner was played short and worked across to Dastan Satpaev, whose drilled shot went through the crowd and was too much for KuPS's goalkeeper Aatu Hakala to deal with.
All the momentum was with Kairat, for whom Edmilson Santos had a couple of decent opportunities, before his side put themselves 2-0 up before the half-hour mark. Jorginho got in between a couple of defenders to prod home Erkin Tapalov's low cross.
The onslaught showed no signs of slowing up, as Kairat had 15 shots in the first half to KuPS' zero, with Edmilson, Jorginho and Valeriy Gromyko all looking lively in front of goal.
Just before the break, the Kazakh champions grabbed their third of the half to go ahead in the tie - Gromyko received a neat through ball from Satpaev, held off the defender and powered the ball into the top corner.
The visitors weathered further pressure in the early stages of the second half before finally creating a genuine chance just after the hour mark, but Mohamed Toure was denied by goalkeeper Aleksandr Zarutskiy.
With 20 minutes to play, Alexander Mrynskiy pounced on a loose ball after an initial attempt was blocked, but Hakala made a good reaction save to keep KuPS in the tie.
His teammates, however, were not able to carve out the chance to stop the tie from slipping away, as Kairat held on for a 3-0 win on the night and a 3-2 aggregate win, and will face the winner of the tie between Slovan Bratislava and Zrinski Mostar in the Third Qualifying Round.
KuPS will now drop down into the UEFA Europa League (UEL), joining the competition in the Third Qualifying Round, where they will meet either RFS or Malmo.