Draper, 22, had come into the match without dropping serve at the tournament so far and he raced into an early 4-1 lead before being pegged back when he was broken for the first time in the seventh game.
But he immediately broke back to love and served the set out, albeit in faltering fashion when he needed a fifth set point to finally seal it.
Left-hander Draper got the key break in the seventh game of the second set and this time clinched the set more emphatically with an ace.
A double-fault at the start of the third summed up the day for world number 74 Van de Zandschulp, who just could not find the same level of consistency he had displayed in his brilliant victory over four-time Grand Slam champion Alcaraz.
Draper had said before this match that he would be trying to be “more aggressive in certain moments” following a series of narrow defeats earlier this year, but he never really needed to be against an opponent who racked up 38 unforced errors and looked to be struggling physically towards the end.
“I think Botic played an incredible match against Carlos [Alcaraz]. It’s not always easy to go on that big stage and play such a player like Carlos and then come out the next round and feel really good because it’s just so tiring being put in that situation, so credit to him,” Draper said.
The Briton, who is at a career-high ranking of 25 and won his first ATP title in June in Stuttgart, now has a chance to reach his first Grand Slam quarter-final in a draw that is shorn of 24-time major champion Novak Djokovic as well as Alcaraz.
It is an opportunity that Draper is relishing after a “tough year” with injuries in 2023 that left him considering his future in the sport.
He reached the US Open fourth round 12 months ago but lost to Russian Andrey Rublev.
“This year, I’ve been able to stay on court to compete all the time on the tour against the best players in the world,” Draper said.
“This is why I put all the hard work in, to play in front of you guys on the biggest stages in the world, so I’m really, really happy and grateful for all the great people around me and we keep wanting more.”