Bertens had been expected to win her ninth match in a row after capturing the Cincinnati Masters title two weeks ago, a run that included victories over the world's top two players Caroline Wozniacki and Simona Halep, but the 103rd-ranked Vondrousova refused to get overawed by the occasion.
The left-handed Vondrousova broke Bertens four times and was dominant in the short rallies, winning 66 to Berten's 44.
The 19-year-old ultimately trampled over Bertens in the third-set tiebreak as the Dutch self destructed by producing four successive unforced errors.
"It was so, so tough for me," Vondrousova said after reaching the fourth round of a major for the first time. "I just stayed calm."
Vondrousova was one of a record six Czech women to have reached the third round of this year's U.S. Open.
Eighth seed Karolina Pliskova is already through to the last 16 and fifth seed Petra Kvitova will also be hoping to increase Czech representation in the fourth round when she faces Aryna Sabalenka later on day six of the championships.
A year after bowing out in the opening round on her Flushing Meadows debut, Vondrousova appears more at ease on the Grand Slam stage.
"I think I'm mentally better. I'm a year older, that's maybe the thing," Vondrousova, who will next face Ukrainian Lesla Tsurenko, said.
"It was very close match this year, so I think I'm just mentally better."
(Reporting By Amy Tennery, editing by Pritha Sarkar)
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