The Spaniard is the highest seed left in the ATP 250 grass-court tournament following David Ferrer's withdrawal through illness, but almost saw his hopes dashed in a 2-6 6-4 6-4 victory over the Frenchman.
After desperately battling to win his opening service game, Verdasco was subsequently broken twice in losing the first set.
However, a single break of serve in each of the following two sets proved enough for the world number 23.
Compatriot and third seed Roberto Bautista Agut also safely negotiated his way into the next round with a 6-1 6-3 win against Daniel Gimeno Traver.
Bautista Agut finished two breaks up in each set to seal a comfortable progression and will now face Australia's Matthew Ebden.
Nicolas Mahut, meanwhile, disposed of Kimmer Coppejans 6-3 3-6 6-3.
Eighth seed Jurgen Melzer will face qualifier Lukas Lacko after a 6-4 6-3 win over Alejandro Falla, while Jan-Lennard Struff was a 6-4 6-2 winner against Lukas Rosol.
Elsewhere, Portugal's Joao Sousa beat Italian Paolo Lorenzi - a late replacement for Ferrer - 6-3 3-6 6-3, and will now face Mate Pavic after the Croatian defeated Alejandro Gonzalez 7-6 (7-4) 6-3.