Boynton's 3-pointer with 3:24 to play gave the Gators a 75-70 advantage and they went on to set up a Southeast Region final matchup against Butler, which will be seeking its second straight Final Four trip. Butler continued its unlikely run with a 61-54 win over Wisconsin Thursday.
Kyle Collinsworth hit a foul shot for BYU to forge a 68-68 tie with 44 seconds left in regulation but he missed his second attempt and neither team could score the rest of the way.
Alex Tyus led Florida with 19 points and 17 rebounds and Chandler Parsons had 16 points, nine rebounds and seven assists.
Boynton added 17 points for the Gators (29-7), who kept their hopes alive for a third NCAA title in six years.
BYU's Jimmer Fredette, the leading scorer in the NCAA this season, produced 32 points in his final college game. Fredette, however, went 11-of-29 from the field and was scoreless during the overtime period. He missed his only two shots in the extra 5 minutes and turned the ball over twice.
The Cougars finished their season at 32-5 and the Mountain West Conference's tremendous campaign came to a close with the elimination Thursday of both the league teams that had made it to the Sweet 16.
San Diego State, the MWC's tournament champion, lost Thursday to Connecticut.
2014: The Year in Fashion [PHOTOS]
UPI Sports Results for September 19, 2014