The Veikkausliiga champions had made it to this stage for the first time by knocking out Moldovan champions Milsami 1-0 on aggregate last week - their first ever two-legged success in the top tier of European club football in qualifiers or competition proper.
while Kairat had got the better of Slovenia's Olimpija Ljubljana, 3-1, to reach this stage for the second time in their history.
KuPS' Mohamed Toure had the first chance of the game, but his effort was too close to visiting goalkeeper, Aleksandr Zarutskiy. Midway through the half, Jorginho had the ball in the net for Kairat, but he was denied the opening goal by the offside flag.
Zarutskiy passed a trickier test as the first 45 minutes was winding down, when Doni Arifi connected with a free-kick delivered into the box, forcing a smart save from the goalkeeper to ensure the teams went in goalless at the break.
But the Kazakh side's goalkeeper could do nothing to stop KuPS taking the lead less than four minutes into the second half.
Joslyn Luyeye-Lutumba burst down the right wing before having the composure to clip the ball to Toure just outside the area. He took a couple of touches to set himself before curling the ball into the top corner.
The Kairat response to going behind was positive, with Valeriy Gromyko and Edmilson Santos testing Johannes Kreidl within the next 10 minutes. But having survived that period of pressure, the home side doubled their lead with 20 minutes to play.
KuPS did well to maintain possession high up the field, before Saku Savolainen on the right side fed Jaakko Oksanen on the edge of the area. Just as Toure had done, Oksanen got the ball out of his feet and curled a beautiful shot beyond the reach of Zarutskiy.
Sadiku was denied again as KuPS looked to all but seal their place in the Third Qualifying Round, but the Kuopio side had to be content with a 2-0 win, especially after Ricardinho missed a late chance that would have changed the complexion of the tie before the second leg.
That second leg will take place in Almaty next Tuesday, when KuPS will be aiming of reaching the third stage of UCL qualification, where they will face either Brann of Norway or Austria's Red Bull Salzburg.