The No. 2 seed from Russia dropped a set for the first time in the tournament but recovered to keep alive his bid for a first Grand Slam title.
"Just want to do a little bit better than last two times and get this last step, which is the toughest one, actually," Medvedev said.
His match Tuesday couldn't have started much easier. He broke six times in van de Zandschulp's first eight service games and won the second set in just 22 minutes.
Van de Zandschulp, a 25-year-old from the Netherlands, was trying to become the first man to come through the qualifying rounds to reach the US Open semifinals but was completely overmatched early, disappointing his orange-shirt-wearing fans inside Arthur Ashe Stadium.
Van de Zandschulp, who had never even been to New York before the US Open, was attempting to become the first man to reach the semifinals in his tournament debut since 1948.
Felix Auger-Aliassime, a 21-year-old Canadian, joined the group of semifinalists later Tuesday, when his opponent, 18-year-old Carlos Alcaraz of Spain, stopped playing in the second set shortly after a visit from a trainer. Auger-Aliassime was coming off two five-set wins in a row and conceded this match while trailing 6-3, 3-1. No. 12 seed Auger-Aliassime will play Medvedev next.
Alcaraz, in his media availability, said his right adductor (thigh) is the reason he retired.
"Before the match I feel it. I feel it," he said. "During the match, I start the match with control the pain. In the end of the first set -- and beginning of the second set -- it started to improve, the pain, little by little."
As in the women's draw, only one man in the quarterfinals already owns a major trophy: Novak Djokovic, who not only is seeking a record-breaking 21st but also trying to become the first man since Rod Laver in 1969 to win a calendar-year Grand Slam.
Medvedev has come close. He lost to Djokovic in this year's Australian Open final and to Rafael Nadal in the 2019 US Open final.
The only way he could meet Djokovic this time would be in the title match on Sunday. But first things first.
"I don't think about him, because as we saw, anybody can beat anybody," Medvedev said. "If he's in the final, and if I'm there, I'm happy. He's also happy, I guess."
The Associated Press contributed to this report.