In a game of desperately few chances and even lower quality, it took almost 80 minutes for a real clear-cut opener.
It came to Daizen Maeda, who was played in one-on-one by Callum McGregor but blasted high over the bar in a shot fitting of the spectacle in Kazakhstan.
After 210 minutes of goalless action, a shootout was required to decide who would qualify, as well as land the £40m prize money. In fitting style, it took another four penalties before the ball hit the back of the net, Aleksandr Martynovich slotting home after Adam Idah and Luke McCowan had spot kicks saved by Anarbekov, and Valeri Gromyko hit the bar with a panenka effort.
Arnie Engels and McGregor scored for Celtic, but Ofri Arad and Egor Sorokin kept Kairat ahead.
Maeda then saw his must-score penalty saved again by Anarbekov to send Almaty wild, create European history.