Woodson completed 32 of 50 passes for the Wildcats (8-5), who also won last year's Music City Bowl against Clemson.
Rafael Little rushed for 152 yards on 28 carries and caught eight passes for 50 yards and a score for the winners. Steve Johnson hauled in seven balls for 124 yards and two touchdowns.
Drew Weatherford threw for 276 yards on 22-of-48 passing with one touchdown and two interceptions for Florida State (7-6), which didn't suit up 36 players caught up in an online test cheating scandal.
Antone Smith rushed for 156 yards on 17 carries and Weatherford ran for two scores for the Seminoles who lost two straight to end the season.
Greg Carr had 99 yards and one touchdown on six receptions for FSU.