Tinkler, 16, scored 14.933 to become only the second British woman to win an individual gymnastics medal, after Beth Tweddle's uneven bars bronze in 2012.
American Biles, 19, scored 15.966 to take the title and add to her all-around, vault and team gold medals.
Biles' US team-mate Alexandra Raisman won silver with 15.500.
"It has been incredible," said Tinkler. "I can't believe it. I'm in total shock.
"I wasn't thinking about winning a medal. My coaches said I had nothing to lose and just to go out and enjoy it. I could hear a lot of cheers from the crowd and that helped me a lot.
"Max Whitlock and Louis Smith have inspired me so much over the years, so to have a medal like them is incredible."
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