He grabbed an offensive rebound on a missed jumper by Scottie Wilbekin and got fouled on the putback. He missed the first free throw, but gathered himself and sank the second for the Gators (6-1).
Florida State (5-2) had a shot to win it, but Ian Miller's desperation heave from half court bounced off the back of the rim, just missing by inches.
Miller finished with 13 points and Boris Bojanovsky had 14 points for Florida State. Aaron Thomas added 12 points and Michael Ojo had 10 for the visitors.
Casey Prather led Florida with 19 points, and Michael Frazier II added 17. Finney-Smith finished with six points and 10 rebounds.
The Gators extended their home-winning streak to 20 and won their fifth straight in the series.
This one was a nail-biter.
Frazier's fifth 3-pointer put Florida up 63-55 with 3 minutes to play. It was the largest lead of the game for either team, but it didn't last.
The Seminoles made eight consecutive free throws, trimming the deficit with the clock stopped.
Wilbekin also helped Florida State get back in the game. Playing in his second game back since a suspension, Wilbekin missed two free throws with 2:18 to play and had a careless turnover on the next possession.
Miller's 3-pointer cut Florida's lead to 65-64 with 1:10 to play.
Bojanovsky's block on Will Yeguete was also key. Bojanovsky, however, fouled on Florida's next possession. Prather made one of his two free throws, but Miller tied it with a pair from the charity stripe with 28.3 seconds to play.
Florida coach Billy Donovan called timeout and drew up a play for Wilbekin. The senior dribbled near midcourt until the clock hit 10 seconds and drove to the elbow. His jumper missed wide, but Finney-Smith was in perfect position for the offensive board.
The Gators finished 15 of 24 from the free-throw line.
FSU was 20 of 23.
It was an ugly, defense-intensive game on both ends.
Florida shot 38.3 percent from the field, getting 32 points in the paint. FSU shot 42.3 percent and scored 26 points in the lane.