Georgia (12-10, 6-4 Southeastern Conference) has won two straight after ending a three-game skid in Thursday's victory over LSU.
Mann, the Bulldogs' leading scorer this season, was just 3 for 8 from the field, but his fastbreak dunk, following a steal by Morris, made it 56-48 with 1:55 left in the game.
Jamal Jones finished with 17 points and was the only player to score in double figures for Texas A&M (13-10, 4-6 SEC). The Aggies have lost six of seven.
Georgia sixth man Nemanja Djurisic, the Bulldogs' second-leading scorer this season, hobbled off the court with 1:42 remaining with an apparent right leg injury.
Georgia's next game is Wednesday at Mississippi State.
Texas A&M took its last lead at 40-39 on Shawn Smith's layup with 10:21 remaining. Bulldogs coach Mark Fox called a full timeout, and Gaines scored 12 of Georgia's next 21 points.
Mann played the game with sore ribs on his right side, an injury sustained against LSU, but he hit four straight free throws in the final 1:09 to help push the lead from seven to 12.
Texas A&M hit just one of 19 attempts beyond the arc. Jones went 1 for 10, hitting a 3 from the right side to cut Georgia's lead to 25-23 at the halftime buzzer.
Donte' Williams had nine points and a game-high 10 rebounds for the Bulldogs.
After Smith's layup, Gaines hit a 3 on Georgia's next possession, and he picked up a loose ball on the Bulldogs' end and jammed to make it 44-40 with 9:19 remaining.
Gaines blocked Jones' runner in the lane to force the Aggies into a shot-clock violation with 5:14 remaining.
Texas A&M had three offensive rebounds on a possession, but couldn't score, and Gaines fed Mann for a fastbreak layup that made it 49-44 with 4:20 remaining.
The Bulldogs went ahead on 52-47 on Gaines' straightaway 3 at the 3:48 mark, and they took a 54-48 lead on an 18-footer by Gaines with 3:01 remaining.
Texas A&M handed out a season-low three assists.