The first set was a tight affair between the competitors as both players were able to grab two break points apiece.
Gasquet, who is seeded ninth for the tournament, had a further eight break points to clinch, but he squandered all of them.
Despite Gonzalez keeping proceedings tight in the tie-break, it was Gasquet who triumphed 9-7 to take the lead in the US.
Gasquet drew on his experience in the second set as he took control of the court by converting two break points and maintaining his serve.
While Gonzalez was able to score four aces, he was forced to succumb to defeat as Gasquet marched into the next phase.