Against Van de Zandschulp, who is enjoying a career-high ranking of 23rd, Medvedev controlled the contest with his serve and allowed his opponent to beat himself.
Zverev won 70 per cent (46-of-66) of his total service points, and while he kept things clean with 17 winners and 17 unforced errors, Van de Zandschulp also posted 17 winners, but with a crippling 38 unforced errors.
There could be another meeting between Zverev and Kyrgios in the quarter-final after Kyrgios collected a straight sets win over Alejandro Davidovich Fokina 7-5 6-4, extending his hot streak to 16 wins from his past 18 matches.
Earlier in the day, world number 12 and one of the favourites for the U.S. Open, Jannik Sinner, needed three hours and 17 minutes to eventually defeat Kyrgios' doubles partner Thanasi Kokkinakis 6-7 (9-11) 6-4 7-6 (8-6).
Sinner will play Serbia's Miomir Kecmanovic next after he defeated last week's Canadian Open champion Pablo Carreno-Busta 1-6 7-5 7-6 (7-4).
It seems like only a matter of time before 19 year-old Carlos Alcaraz lifts his first grand slam trophy, and his run to the U.S. Open is back on track after his first back-to-back losses of the year, comfortably handling the challenge of Mackenzie McDonald 6-3 6-2.
Tommy Paul won his all-American battle against Jenson Brooksby 6-3 6-2, while Australia's Alex de Minaur was similarly impressive in his 6-2 6-2 victory against Henri Laaksonen in just over an hour.
Italy's Fabio Fognini will play world number 11 Andrey Rublev in the second round after defeating Albert Ramos-Vinolas 6-2 6-3, and world number 13 Taylor Fritz could find himself in the top-10 for the first time in his career with a few more wins this week after a dominant 6-1 6-1 smashing of Sebastian Baez.
American wildcard Ben Shelton pulled off a 7-6 (7-5) 3-6 7-5 upset of Italy's Lorenzo Sonego, and Croatia's Filip Krajinovic was too good for England's Daniel Evans on the way to a 6-3 6-4 triumph.
Marin Cilic, the 14th seed, beat rising Finnish talent Emil Ruusuvuori 6-4 6-7 (5-7) 7-5, and in the final match of the night, 13th seed Diego Schwartzman beat Russia's Aslan Karatsev 7-6 (7-3) 3-6 6-2.