Stepanek converted the first match point with a serve winner.
Bozoljac said he didn't shake hands with Stepanek after the match.
Stepanek called it "a gesture. It shows his weakness."
The Czechs need one more point to reach their third semifinals in four years and to avenge a 3-2 loss to Serbia in the 2010 semifinals.
In Sunday's reverse singles, Berdych faces Janko Tipsarevic in a matchup of two top-10 players.
The winner faces Argentina or Croatia in September in the semifinals.
Spain 2, Austria 1
Austria reduced Spain's lead to 2-1 by winning the doubles in the Davis Cup quarterfinals on Saturday.
After squandering four match points, Peya hit a winner at the net to end the match.
The decisive reverse singles will be on Sunday.
The five-time defending champions, playing without Rafael Nadal, will again look to David Ferrer and Nicolas Almagro, who brushed aside Jurgen Melzer and Andreas Haider-Maurer in Friday's opening singles.