The 24-year-old, who withdrew from the women’s team competition last Tuesday due to mental health concerns, finished behind Chinese teenagers Chenchen Guan and Tang Xijing.
The bronze is her seventh Olympic medal.
Competing second, Biles had been ranked in second place behind Tang up until Guan delivered a routine which scored 14.633 to take gold.
Ironically, Biles’ bronze medal-winning total of 14.0 was lower than her heat score of 14.066, with a series of minor wobbles.
Biles looked pleased and relaxed as she waved to the crowd on Tuesday after struggling with spatial awareness issues, known as the 'twisties', just a week before.
After withdrawing from the team final last week, Biles pulled out of the first three of the four individual finals for which she had qualified.
The US star been expected to equal or surpass her historic quadruple gold medal haul from Rio in 2016.
But she left the floor after one rotation of the team final after failing to land a vault.
Biles later explained on social media that she had been struggling with the 'twisties': “It has never transferred to bars and beam for me. This time it’s literally in every event which sucks.
“(You) literally cannot tell up from down. It’s the craziest feeling ever, not having an inch of control over your body.”
Although the beam is not considered to be Biles’ greatest strength, she has won three world titles and a bronze medal in Rio 2016.