Zverev, seeded second, has failed to go beyond the fourth round in his past nine visits to the All England Club.
But the 29-year-old German is on course to end that frustrating streak this year.
Fresh from finally winning his maiden Grand Slam title on clay at Roland Garros in June, Zverev is looking more comfortable than ever on the grass courts of south-west London.
He dispatched 92nd-ranked Giron 6-2, 7-6(4), 6-4 on Court One and will face Czech 13th seed Jiri Lehecka in the last 16.
"He plays fantastic on grass. I knew I had to be 100 per cent ready from the first point and I'm very happy to be through in straight sets," Zverev said.
"Winning in Paris, coming here with a title, it helps. It was my first one and I want to keep going.
"I feel different than last year. I feel like I play better tennis and have more confidence in important moments. I want to keep going and play more important matches."