Hajek opened the series with a 6-3, 6-2, 6-4 victory over Mikhail Kukushkin and Rosol followed with a 4-6, 6-4, 6-2, 7-6 (8-6) decision over Andrey Golubev.
That puts the Czechs, who won the Davis Cup title last year, within one win from advancing to the tournament's Sept. 13-15 semifinals. Whichever team advances out of Kazakhstan will go against the winner of a series between Argentina and France in the semis.
Hajek faced just one break point -- in the day's third game -- and cashed in half of his break opportunities in his win. He won 75 percent of the points on serve.
Rosol dropped serve twice. A break in the first set dropped him behind Golubev but three breaks over the next two sets, while losing just seven points on serve, put Rosol up 2-1 in the match.
Golubev got his second break of the day early in the fourth set but Rosol evened the set in the 10th game and was able to finish off his second match-point chance.
Sunday's schedule calls for reverse singles, with Rosol meeting Kukushkin in the opener and Hajek to play Golubev in the series' final match.