Verdasco, the tennis tournament's top seed, downed Frenchman Mathieu Montcourt 7-6 (7-5), 6-1. The third-seeded Moya rallied to beat Serbian Viktor Troicki 3-6, 7-6 (7-2), 7-5.
Verdasco converted six of 12 break points and the inexperienced Montcourt committed seven double-faults in their one hour, 28-minute match.
Moya's victory stopped his three-match personal losing streak.
In other matches, Italian Fabio Fognini topped compatriot and two-time Umag finalist Filippo Volandri 6-4, 6-4; and Germany's Mischa Zverev knocked off No. 8 seed Marcos Daniel of Brazil 6-4, 6-2.
The second round concludes Thursday with matches involving Croatian crowd favorites Ivo Karlovic and Ivan Ljubicic, the second and fifth seeds, respectively.
When the tournament concludes with Sunday's title match, the 2008 Umag champion will earn an $86,000 first-place check.