The Gators (26-2, 15-0 SEC) nearly squandered an eight-point lead over the final 5 1/2 minutes, allowing the Commodores to pull within a point, 54-53, on a pair of Kyle Fuller free throws with 1 minute left.
But Finney-Smith (19 points, nine rebounds) connected from long range on the next possession, putting Florida ahead by five.
Fuller responded with a layup and the Gators' Scottie Wilbekin managed to hit just 1-of-2 free throws with 10 seconds to go, giving the Commodores one last chance.
Fuller, however, was off the mark on a 3-point try, allowing the Florida to extend the best start in program history.
Patric Young added 12 points and seven rebounds.
Vanderbilt (15-12, 7-8) was led by Rod Odom with 12 points.
Rosie O'Donnell unveils nearly 50-pound weight loss
Iranian woman stops the execution of son's killer