Fredette, the nation's 12th-leading scorer, made a 3 from the top of the key to put the seventh-seeded Cougars (30-5) ahead 90-83 with 2:42 left, then added another with just over a minute remaining to put the game away.
Michael Loyd Jr. added a career-high 26 points, pushing BYU to a lead it would hold until a late second-half rally by Florida. Loyd scored all the Cougars' points in the first overtime.
Kenny Boynton scored 27 points, and rallied the Gators (21-13) from a 13-point second-half deficit in their first NCAA tournament game since winning back-to-back national championships in 2006 and 2007.
BYU snapped a streak of seven straight first-round losses in the tournament.