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German star and second seed Zverev accounted for teenager Carlos Alcaraz, the 2019 runner-up winning the opening-round clash 6-3 6-1.
Alcaraz – in his first showdown against a top-10 player – broke for a 2-0 lead and while he continued to go toe-to-toe with Zverev through six games, the 17-year-old was unable to maintain his charge.
Zverev broke for a 4-3 lead and the 2020 US Open finalist did not look back, winning nine of the last 10 games to advance in 84 minutes.
"He started off playing extremely well, he didn't miss a lot of balls. Then I felt more comfortable on court, this is the first round," Zverev said.
"It's not always easy to play a first round, especially against a player like this, who is very young, who is very motivated and who wants to compete. I'm actually quite happy with how the match went."
Next up for Zverev – who reached the Australian Open quarter-finals in February – is Serbia's Laslo Djere.
Djere took down Steve Johnson 6-2 6-3 for a spot in the last 16.
Meanwhile, Korda capped his Acapulco bow with a shock 7-5 6-2 win against former world number three Marin Cilic.
Korda claimed his first ATP 500-level victory having saved 10 of the 11 break points he faced in the opening set against the 2014 US Open champion.
Canadian sensation and seventh seed Felix Auger-Aliassime awaits after the 20 year-old converted his sixth match point to see off Tennys Sandgren 6-3 7-6 (10-8).
Elsewhere, John Isner and Dominik Koepfer also advanced.