Stepanek survived a tight 4-6, 6-3, 6-3, 3-6, 6-3 decision over Andreas Seppi before Berdych took out Simone Bolelli 6-3, 4-6, 7-5, 6-2.
That left the Czechs needing just one win in Saturday's doubles match or either reverse singles match Sunday to earn a spot in the Davis Cup quarterfinals April 6-8. The Czech-Italy winner will face either Serbia or Sweden in the next round. Serbia is up 2-0 in that series.
Berdych, No. 7 in the world, had 17 aces and 57 winners against 26 unforced errors in the win over Bolelli. He won 80 percent of the points on serve, a figure enhanced by a fourth-set effort in which he won 16-of-18 service points. He closed the match with a hold at love.
Stepanek won just 10 more points (159-149) than Seppi in the match but nearly all of that advantage came in the fifth set when he lost five points on serve.
Daniele Bracciali and Potito Starace get the call to keep Italy's hopes alive in Saturday's doubles. The Czechs counter with Frantisek Cermak and Lukas Rosol.
Berdych has the first match Sunday, going against Seppi, while Stepanek and Bolelli set scheduled for the fifth match of the series.