Rosol beat Yasutaka Uchiyama 6-3, 3-6, 6-4 and Vesely overpowered Taro Daniel 6-4, 6-4 to complete the one-sided tie.
The Czech team secured a berth in the semifinals on Saturday, beating Japan in the doubles match to give the defending champions an insurmountable 3-0 lead.
Both teams were without their top players. Tomas Berdych sat out the tie following two tough tournaments in the United States, while Kei Nishikori was sidelined after hurting his groin in the Sony Open in Miami last week.
The Czech Republic plays the winner of the quarterfinal match between France and Germany in the last four in September.