World number three and 2022 champion Alcaraz lost 6-1, 7-5, 6-4, ending his bid to become only the third man in the modern era to win the French Open, Wimbledon and US Open in the same season.
Alcaraz had made at least the quarter-finals in his three previous appearances in New York and Thursday's defeat was his earliest at a Grand Slam since a second-round exit at Wimbledon in 2021.
"I'm a little lost for words," said van de Zandschulp after only his second career win over a top-five player.
"It was an incredible evening out here in my first time in the night session on Arthur Ashe. I had some unbelievable points at the net.
"I tried to stay calm. You have to keep your head against these guys otherwise they will take advantage."
Alcaraz, who had needed four sets to see off Australian qualifier Li Tu in the first round, appeared on court with his left thigh heavily strapped.
That proved the least of his problems as he dropped serve twice in the opening set, failing to hit a single winner against his 28-year-old opponent, who made the quarter-finals on his debut in 2021.
Alcaraz squandered break points in the second game of the second set and the Dutchman made him pay with a break of his own for a 2-1 lead in front of a stunned crowd.
The 2022 champion hit back immediately to level at 2-2 but it was a brief respite from his troubles as his inspired opponent broke again for 6-5 and took the set when Alcaraz fired a service return wide.
The 21-year-old had never had to come back from a two sets deficit in his young career and went into Thursday's match with a 2-0 head-to-head edge over the Dutchman.
However, he slipped 2-3 down, recovered to 3-3, greeting the rare chink of light with a wide grin, but was then broken again in the ninth game.
Van de Zandschulp was not to be denied and claimed victory when Alcaraz fired his 27th and final unforced error of the contest.
He becomes the first Dutchman to defeat a top three player at a Grand Slam since Richard Krajicek beat number one Pete Sampras in the quarter-finals at Wimbledon in 1996.
He is the first from his country to achieve it at the US Open since his coach Paul Haarhuis knocked out Boris Becker in the third round in 1991.
