Medvedev had been building momentum ahead of the year's last Grand Slam at Flushing Meadows by winning a tune-up event in Los Cabos without dropping a set but was beaten by Nick Kyrgios (28) in the second round of the Canadian Masters in Montreal last week.
The Russian started strongly against Dutchman Van de Zandschulp with an immediate break before going on to win the first set.
The second set posed more of a challenge, with the world number one having to save a set point, but he capitalised on several double-faults from Van de Zandschulp before wrapping up the match.
"The second set was going not that bad," Medvedev said in an on-court interview. "He was not serving especially well today, doing a lot of double faults. So I tried to keep pressure, use my chances.
"Two breaks on these fast courts is enough to win and I should have done a little bit better on my serve. But the first match is tricky and I'm happy that I managed to go through, and hopefully I will raise my level in the next matches."
Medvedev will face either American Tommy Paul (25) or Canada's Denis Shapovalov (23) in the next round.
In other first-round action at the Masters 1000 tournament, Paul beat compatriot Jenson Brooksby (21) 6-3 6-2, Australia's Alex de Minaur (23) dispatched Swiss Henri Laaksonen (30) 6-2 6-2, Croatian Borna Coric (25) defeated Lorenzo Musetti (20) 7-6(2) 6-3 and Spaniard Albert Ramos-Vinolas (34) beat Italy's Fabio Fognini (35) 6-2 6-3.
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