KuPS made the long journey to the Caspian Sea after seeing off Vardar of North Macedonia in the previous round, and are now tasked with beating the champions of Azerbaijan back in Kuopio in the second leg on 28th July if they are to remain in the UCL.
Sabah were looking the more likely to break the deadlock - Veljko Simic went close with an early chance before VAR came to KuPS' rescue in the 33rd minute, disallowing an Umarali Rakhmonaliyev goal for a foul in the build-up.
Johannes Kreidl in the visitors' goal was tested by Joy-Lance Mickels late in the half, before the breakthrough arrived in first-half added time, as a selfless Ivan Lepinjica played in Simic, who guided the ball past Kreidl to give the hosts a deserved lead.
Rakhmonaliyev went close again in the second half, and despite KuPS generating just one effort on goal over the 90 minutes, they nearly made Sabah pay for not pressing home their advantage when the ball fell to Jaime Moreno in stoppage time, but he sent his effort a whisker over the crossbar.