Berdych needed 3 hours, 45 minutes to grind out a 7-6 (8-6), 7-6 (8-6), 7-6 (9-7) victory over the Serb, overcoming 62 unforced errors and firing 12 aces along the way.
His victory gave the Czechs an insurmountable 3-1 lead in the best-of-five tie, earning them their third Davis Cup semifinal appearance in four years.
The doubles team of Berdych and Radek Stepanek had put the Czechs in the driver's seat Saturday with a 6-4, 6-2, 7-6 (7-4) win over the Serbian team of Nenad Zimonjic and Ilija Bozoljac.
The Czechs now go on to face Argentina in September's semifinals.
Argentina advanced by defeating Croatia on Sunday.