Berdych and Stepanek allowed the Serbia team of Ilija Bozoljac and Nenad Zimonjic just one break point during a 6-2, 6-4, 7-6 (7-4) win.
That combined with Berdych's singles win Friday gives the Czech a 2-1 lead in the best-of-five series. The Czechs are seeking a second consecutive Davis Cup title.
Serbia needs to win both singles matches Sunday to collect its second Davis Cup championship.
World No. 2 Novak Djokovic goes against Berdych, seventh in the rankings, in the first match Sunday while Stepanek is to take on Dusan Lajovic. Djokovic, who beat Stepanek in the series opener Friday, owns a 14-2 record versus Berdych. Lajovic and Stepanek have never before played each other.
It was little surprise that Czech interim captain Vladimir Safarik turned to Berdych and Stepanek over the announced team of Jan Hajek and Lukas Rosol for doubles. The Czechs often make that move and have been rewarded: Saturday's win gives the pair a 6-0 record in Davis Cup play the last two years.
There was a surprise when the Serbs, however, didn't put Djokovic on the court with Zimonjic. Instead, the No. 2 player in the world was regulated to watching on the sidelines as Berdych and Stepanek defeated Bozoljac and Zimonjic.
The Czechs opened the match with a break and added another in the seventh game before Stepanek held serve at love to close out the first set. The second set also began with a break by Berdych and Stepanek that they made stand for a 2-0 set lead.
Bozoljac and Zimonjic finally earned a break point in the sixth game of the third set but Stepanek was able to hold. The Serbs then led 3-1 in the tiebreaker only to have the Czechs run off six of the next seven points to end the match.