On a scorching Court Philippe Chatrier, Zverev needed just over two hours to see off world number 95 Bonzi on the opening day of the clay-court Grand Slam.
Zverev broke in the sixth game of the first set to take control.
The pair exchanged breaks midway through the second set, before Zverev pounced on his opponent's serve to get ahead 5-4, immediately sealing the two-set lead with a clean hold.
With the mercury tipping past 30 °C, Zverev raced through the final set to book his ticket into the round of 64.
"Very good start to the tournament, always nice to start with a straight-sets win," Zverev said on court.
"(It was) all around a good performance, very happy with the start."
The 29-year-old German second seed will next face Czech Tomas Machac, who earlier beat Belgium's Zizou Bergs 6-4, 6-4, 6-3.
Karen Khachanov of Russia also advanced into the second round with a straight-sets win over French wildcard Arthur Gea.
Czech 26th seed Jakub Mensik made similarly short work of another French invitee, Titouan Droguet, as the 20-year-old strolled through 6-3, 6-2, 6-4.
Brazilian 19-year-old Joao Fonseca then put on a clinical display to see off hometown qualifier Luka Pavlovic without dropping a set.
Fritz dumped out by Basavareddy
In the first big shock of the Slam on the men's side, seventh seed Taylor Fritz was stunned by wild card and fellow American, Nishesh Basavareddy. The world number 156 clinched the tight contest 7-6(5), 7-6(5), 6-7(9), 6-1.