The three-time Grand Slam runner-up won through 6-2, 7-6(4), 6-1 in two hours and 30 minutes, and next plays 35th-ranked Dutch player Tallon Griekspoor.
"It's a tournament I look forward to the most during the year," said Zverev after his match on Court Philippe Chatrier.
"Especially this court is so special to me. I've had some of the best and worst memories of my life on this court.
"The crowd always supports me. The love story continues."
Zverev has reached at least the semi-finals on each of his last four appearances at Roland Garros.
"He (Cobolli) probably should have won the second set, but I kept fighting and then I had a bit of a mental advantage and it went my way."
Griekspoor battled past American Ethan Quinn 4-6, 6-1, 6-7(2), 6-1, 6-4 in three hours and 16 minutes.