Giron lost the first set in a tie-break, however, the American fought back in an encounter lasting nearly three hours on Wednesday (AEST).
Twenty minutes of match time came in the final game, which encompassed all of Giron's match points.
Munar successfully staved off the first nine and had seven break points of his own. Yet his resistance was finally broken as Giron prevailed at the ATP Tour tournament.
Next up for Giron is a meeting with Australian third seed Alex de Minaur.
Eighth seed John Millman enjoyed a more routine 6-2 6-4 victory over Mikael Ymer, while there were also victories for Laslo Djere, Dimitar Kuzmanov and Egor Gerasimov.