Biles – a gold medallist in the team, all-around and vault in Rio – fought to stay on the apparatus during her routine, scoring just 14.733 points.
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Wevers, 24, took gold with 15.466 points ahead of American Lauren Hernandez, who won silver with 15.333.
The Dutch gymnast finished runner-up to three-time world all-around champion Biles on the beam at last year's world championships.
Despite getting off to a strong start the 1.45m (4ft 9ins) Texan put her hands down on the beam after landing a front somersault.
Biles still has a chance for a record-equalling fourth gold in Tuesday's floor final.
Soviet gymnast Larissa Latynina (1956), Czech Vera Caslavska (1968) and Romanian Ecaterina Szabo (1984), have all also won four gold medals at the same Olympics.
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