The pair defeated the Kazakh team of Evgeny Korolev and Yuriy Schukin 6-4, 6-4, 7-6 (7-4), building on Berdych's singles win Friday over Mikhail Kukushkin.
Berdych, the world's seventh-ranked player, pulled the Czechs even with the victory after Kazakhstan had opened a 1-0 lead behind Andrey Golubev's five-set win over Jan Hajek.
Berdych will try to clinch the round for the Czech Republic Sunday when he faces Golubev in the day's first reverse-singles contest.
The winner will go on to face Argentina in the quarterfinals. Argentina advanced Saturday with a doubles win over Romania.
Celebrity Couples of 2014 [PHOTOS]