Backup quarterback Jacoby Brissett tied the score with 1:42 remaining with a 3-yard touchdown pass to Quinton Dunbar and Florida's defense forced the Ragin' Cajuns to punt on the ensuing possession with 13 seconds left.
But Loucheiz Purifoy blocked the attempt and the ball fell into the arms of Jenkins, who raced in for the decisive touchdown.
Starting quarterback Jeff Driskel had left with an ankle injury earlier in the second half.
Despite the win, the Gators (9-1) lost out on a chance to clinch the Southeastern Conference's East Division when No. 5 Georgia defeated Auburn.
Terrance Broadway went 16-of-23 for 171 passing yards for Louisiana-Lafayette (5-4).