Georgia faced a 10-0 deficit after the first quarter, but its defense stiffened and held Kentucky to three points in the game's final 45 minutes.
Brown gained 73 yards on 22 attempts. Koshown Moreno scored once on 22 rushes for 130 yards.
Quarterback Matthew Stafford was just 12-for-22 for 98 yards and two interceptions passing but he ran for a touchdown in the second half.
Brandon Coutu's 46-yard field goal with just over two minutes remaining effectively sealed the win for the Bulldogs (9-2, 6-2 SEC), who overcame four turnovers to earn at least a share of the SEC East Division crown.
Kentucky quarterback Andre Woodson completed 24 of 42 passes for 267 yards with one touchdown and one interception. Lones Seiber's 32-yard and 44-yard field goals accounted for the rest of Kentucky's scoring.
Wide receivers Steve Johnson and Jacob Tamme combined to make 14 receptions for 168 yards, while Kennan Burton caught a 36-yard first-quarter touchdown pass for the Wildcats (7-4, 3-4), who haven't won in Athens since 1977.