Hajek defeated Mikhail Kukushkin 6-3, 6-2, 6-4 before Rosol rallied to beat Andrey Golubev 4-6, 6-4, 6-2, 7-6 (6) in the second singles.
It was the first loss at home for Kukushkin, who has a 12-6 record in the competition.
The Czech Republic is without Tomas Berdych, who has a shoulder injury. Their No. 2 player Radek Stepanek, who had a neck operation in January, was rested for the opening singles. He's scheduled to play doubles on Saturday and may step in for a reverse singles match on Sunday.
The Czech Republic beat Spain in the final last year for its first title as an independent nation. Kazakhstan is looking for its first semifinal appearance.
The winner of the best-of-five series will face either Argentina or France in the semifinals in September.