The German second seed will face home player Quentin Halys on Friday for a place in the second week at Roland Garros after a 6-4, 6-2, 6-2 win in the night session match on Court Philippe Chatrier.
"I'm happy where I am, I'm feeling confident in my game and moving forward, looking forward to hopefully being back on this court," said Zverev.
Machac, the world number 43, struggled with his movement and took a lengthy medical time-out in the second set for treatment to his left foot.
"I thought for one and a half sets it was an amazing level... unfortunately, the injury stopped the match flow a little bit," added Zverev. "I'm happy to finish it in three sets."
Zverev took full advantage with a dominant display, saving the only break point he faced as he dismantled Machac in just an hour and 48 minutes.
The world number three has come closest to breaking his major duck in Paris, reaching at least the quarter-finals in each of the past five years, suffering a five-set loss to Carlos Alcaraz in the 2024 final.