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 stunned by Basavareddy
Later in the day, seventh seed Taylor Fritz was stunned 7-6(5), 7-6(5), 6-7(9), 6-1 by fellow American Nishesh Basavareddy in the first round of the French Open on Sunday as his injury-disrupted claycourt season ended with a whimper.
Fritz returned with an opening defeat at Geneva last week after two months away due to a knee injury, and he showed signs of rust under the afternoon sun on Court Suzanne Lenglen against 21-year-old wildcard Basavareddy.
After a tight opening set slipped away in a see-sawing tiebreak, the 28-year-old came out firing in the second frame, only to surrender an early break as Basavareddy mixed up his game with impeccable drop shots to trouble his compatriot.
Fritz was soon staring at back-to-back first-round defeats at Roland Garros when world number 148 Basavareddy unleashed a powerful serve late in the second-set tiebreak to double his lead in the match.
The third set went with serve for 12 games, and Fritz let out a loud scream when he won the tiebreak after saving a match point, but Basavareddy was not to be denied a maiden top-10 win, and he took complete control of the fourth set and finished the job.
Up next for Basavareddy is a second-round clash with either Alexander Shevchenko or Alex Michelsen.