Biles, 19, had been tipped to scoop a record five golds at Rio, but lost her footing after a somersault and had to grab the beam with both hands to save herself from falling off.
She still claimed bronze, but it was 24-year-old Wevers who won the top prize - the Netherlands' first women's individual gymnastics gold.
Biles' fellow American, Laurie Hernandez, won silver.
Biles, a double world champion on the beam, will be back tomorrow morning when she will be favourite to win the floor exercise.
She has already won gold in the team competition, individual all-around competition, and on the vault.