In the doubles on Saturday, Ivan Dodig and Nikola Mektic, who are 1-0 together in Davis Cup, will play Kazakhstan's Timur Khabibulin and Aleksandr Nedovyesov, who are 2-0.
The 92nd-ranked Kukushkin was a set down and a break down twice in the second set, but he recovered immediately each time and won a smart tiebreak 7-5 on his third set point. He then broke Coric at 2-1 in the third, prompting his 28th-ranked opponent to smash his racket into the clay.
Kukushkin finally broke Coric's resistance when he broke again for 2-1 in the fourth and served out.
Former U.S. Open champion Cilic hit 10 aces and had 32 winners to Popko's 12. He lost only 11 points on serve in a first-time matchup between players ranked No. 3 and No. 258.
Croatia is trying to reach the semifinals for the second time in three years, while Kazakhstan is trying to reach the semifinals for the first time.