The 26-year-old Granollers overcame resistance from the Russian-born Zverev, who is ranked 190th.
Zverev held his serve and saved six break points before Granollers finally broke him in the ninth game to take a 5-4 lead to take the first set. The 22nd-ranked Spaniard broke the German twice in the second to wrap up the match.
Viktor Troicki stopped Benjamin Becker 6-2, 7-6 (6) to set up a second round meeting with third-seeded Tomas Berdych.
Earlier, Florian Mayer beat American qualifier Tim Smyczek in straight sets to set up a second round meeting with five-time winner Roger Federer on Thursday.
The same day, Rafael Nadal will open against Lukas Lacko, who beat Konstantin Kravchuk 2-6, 6-2, 6-3 in the first round.
Nadal won a record seventh French Open championship on Monday after defeating top-ranked Novak Djokovic at Rolland Garros.
- Marcel Granollers
- Mischa Zverev
- Bernard Tomic