The 42nd-ranked Czech saved seven break points while converting his one chance, and took nearly two hours to put away the No. 3 seed.
Rosol, who lost in the final in Stuttgart against Roberto Bautista Agut on Sunday, will next play Philipp Kohlschreiber, the seventh-seeded German who defeated 2011 champion Gilles Simon of France, 7-5, 6-3.
The 83rd-ranked Kamke next faces 17-year-old Alexander Zverev, another Hamburg native, who defeated Santiago Giraldo of Colombia 6-4, 7-6 (6) to keep his breakthrough run going.
The 285th-ranked Zverev defeated No. 5 seed Mikhail Youzhny of Russia on Wednesday.