The 23-year-old Briton broke his Czech opponent four times on his way to a 6-4 7-5 victory and will now face Italian fifth-seed Fabio Fognini in the first round.
In other qualifying matches, Germany's Andreas Beck beat countryman Dennis Bloemke 6-3 6-1 while another German, Nils Langer, beat Kazakh Evgeny Korolev 6-4 6-3.
Spaniard Daniel Munoz-De La Nava is also into the main draw after beating Poland's Grzegorz Panfil 7-6 (7/2) 6-1.
German wildcard Michael Berrer was another winner as he advanced to the second round with a 6-4 3-6 6-1 win over Spain's Daniel Gimeno-Traver.
Eighth seed Martin Klizan advanced to the second round with a 6-2 6-3 win over Albert Montanes while Tobias Kamke could not continue the theme of German success as he was beaten 6-3 7-6 (7/4) by Roberto Bautista Agut.