Mann sank 8-of-11 shots from the floor and 12-of-13 at the charity stripe for the Bulldogs (20-13), who will face Louisiana Tech in the second round. Marcus Thornton added 16 points in the win.
Sandro Carissimo paced Vermont (22-11) with 11 points and five rebounds, while Candon Rusin chipped in 10 points. Brian Voelkel handed out six assists and swiped six steals for the visitors.
Georgia knocked down 55.6 percent of its field goal tries and limited the Catamounts to just 33.3 percent efficiency from the floor in the first half. Vermont went into the intermission trailing, 29-20, despite forcing the hosts into 13 turnovers before the break.
Vermont heated up to make good on 63.6 percent of its shots from the floor in the second period. The Bulldogs extended their lead as they converted 17-of-18 attempts at the free-throw line over the final 20 minutes of the contest.
Georgia enjoyed a 31-18 advantage in the rebounding battle.