The top seed swept aside the Dutchman with a 6-2 7-5 6-1 win in the night session on Arthur Ashe Stadium.
Following his victory, Zverev revealed he had stayed up to watch Shelton's win over defending champion Alcaraz in a match that ended at 03:34 local time on Wednesday - the
"I watched Shelton-Alcaraz until 3.30am last night," Zverev said.
"I did not go to bed, I was up late, I was watching TV because I was sleeping in until 1pm anyway. "
The 29-year-old has had plenty of his own late-night finishes at this year's tournament, with his opening three matches ending at 1:55am, 2:15am and 1:09am.
Bidding to reach a third successive Grand Slam final, Zverev will face Russian Karen Khachanov, who advanced after Alexander Blockx retired trailing 6-2 7-5 3-2.
With his win over Van de Zandschulp, Zverev has reached all four Grand Slam semi-finals in a single season for the first time in his career.
"I think the more often that you get to that stage, the more normal it becomes, which is something that you should never take for granted," said Zverev, who ended his wait for a maiden major at this year's French Open.
"I think it's a very special feeling when it becomes normality. "
Playing in his first Grand Slam quarter-final, the 21-year-old first received medical treatment for a blister on his foot before struggling with a back issue.
"I think today my body just gave up," said Blockx, who had come close to retiring from his fourth-round win against Francisco Cerundolo with a rib injury.
"During the match, it was quite funny to me. I just had to laugh it off because it was just something new the whole time.
"I felt it was getting worse and worse. I couldn't land without feeling pain, without feeling my back, my ribs. So I just didn't want to yeah, maybe make it worse to miss the entire season. "
Khachanov wished Blockx a "speedy recovery" after he secured his place in a Grand Slam semi-final for the first time since the Australian Open in 2023.
A former world number eight, a disappointing run of results over the past few months have resulted in Khachanov's ranking dipping to 50 and he is unseeded at a major for the first time since 2018.
The 30-year-old said he will have to come up with his "best form" to defeat Zverev and reach his first major final.
"He's been in the top five for so many years. This year he did also step forward. He won his first Grand Slam, played the final in Wimbledon, as well. He's coming up really in hot form," Khachanov said.
Zverev, meanwhile, says he will have to be wary of his opponent and his extensive knowledge of the game.
"We call him [Khachanov] 'The Professor', actually, because he knows everything about tennis, and he knows every single string tension, knows what everybody does, so that's quite fun," Zverev said.
