Lukas Rosol beat Evgeny Korolev 7-6, 67, 7-6, 6-2 to give the Czech Republic an unassailable 3-1 lead over Kazakhstan and propel the defending champions into the Davis Cup semi-finals on Sunday.
Rosol battled Korolev, who had replaced Kazakh number one Mikhail Kukushkin for the first reverse singles, for more than three-and-a-half hours before sealing the quarter-final tie, played on clay at the National Indoor Tennis Centre in Astana.
The Czechs had built up a 2-0 lead after winning both of their singles matches on Friday but the home team avoided an early exit by taking Saturday's doubles.
The visitors were missing their top player, world number six Tomas Berdych, who pulled out of the tie after injuring his shoulder at last week's Sony Open in Miami.
In September's semi-finals, the Czechs face Argentina or France. Argentina led their quarter-final 2-1 going into the final day in Buenos Aires on Sunday.