World number two Zverev, who bagged his maiden slam at the French Open in June, never looked in trouble as he powered to a 6-2, 7-5, 6-1 victory at the Arthur Ashe Stadium.
The 29-year-old German star, a losing finalist at the US Open in 2020, will face unseeded Russian Karen Khachanov in the semi-finals on Friday.
Zverev will head into the last four brimming with confidence after an assured performance against the 70th-ranked van de Zandschulp, who was broken twice in the opening set as Zverev took control.
"I think you saw in the first two rounds, I really didn't play well," Zverev said.
"In other tournaments I would be out in the first round, but here I can fight back and play best of five sets, play 4-1/2 hours. I've started playing better throughout the tournament, and I'm definitely happy to be in the semi-finals here."
Van de Zandschulp did show some fight in the second, and threatened to level when earning a set point on Zverev while leading 5-4.
But the Dutchman botched a return to let that chance slip, and Zverev leveled at 5-5 before breaking in the next game for a 6-5 lead when van de Zandschulp sent a wild forehand wide.
Zverev held serve to take the set and a 2-0 lead, and van de Zandschulp's fleeting resistance was broken.
The 30-year-old quickly went on a double break down to trail 4-1, and then was broken again in the seventh game as Zverev wrapped up victory.
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