John Egbunu chipped in 16 for Florida, including all four of his free throws in the final 2 minutes.
The Gators (12-6, 4-2 Southeastern Conference) overcame an 11-point deficit in the second half and won back-to-back league games for the first time this season.
It was Finney-Smith's fourth double-double in the last five games. Allen, a freshman, topped his season average (11 points) for the sixth time in the last eight outings. Egbunu was even more clutch, scoring six points in the final 3 minutes of a back-and-forth game.
The Bulldogs (7-10, 0-5) lost another close one and fell to 0-5 in league play for the first time since 1986.
The Bulldogs had all the momentum down the stretch, but were doomed by bad shots -- Aric Holman's missed 3-pointer with a little more than a minute to play and then Travis Daniels' missed jumper in the final minute. Despite those, Mississippi State took the lead on Sword's driving layup with 48 seconds remaining, but Florida went back in front when Egbunu hit two more from the charity stripe.
Still, State had a final chance.
Florida's Justin Leon missed two free throws, keeping it 79-78 with 18 seconds to play. Ready dribbled across midcourt, started to drive into the lane, pulled up from about 12 feet and hit the back of the rim. Florida came down with the rebound, and Allen sank both on the other end to put the Gators up 81-78.
Quinndary Weatherspoon threw the long inbound pass out of bounds, and Florida started celebrating.
Mississippi State's Gavin Ware, the team's leading scorer, fouled out in the final minutes with 5 points.
Mississippi State: Ware had scored in double figures in each of the last 16 games. ... Hasn't won in the O'Connell Center since 2008.
Mississippi State hosts rival Mississippi on Saturday.
Florida hosts Auburn on Saturday.