The Bulldogs (16-11, 10-5 Southeastern Conference) have won six of seven.
The Tigers, coming off Saturday's dispirited loss at Alabama, might have hurt their chance of making a sixth straight appearance in the NCAA tournament.
Georgia, which went just 6-6 in nonconference play, began the SEC schedule by snapping Missouri's 26-game home winning streak on Jan. 8, is alone in third place in the SEC behind No. 1 Florida and No. 17 Kentucky.
Missouri dropped into a two-way tie for 10th place with Vanderbilt.
The Tigers' last lead came on Brown's layup midway through the first half. Missouri was outscored 16-6 over the final 9:12 before intermission.
Mann opened up the second half with a straightaway 3, and Juwan Parker's three-point play a couple of minutes later gave Georgia its first double-digit lead at 37-26.
Despite outrebounding the Bulldogs, Missouri struggled in nearly every other facet of the game, missing 15 of its first 16 attempts beyond the arc, getting outscored 38-18 in the paint and 11-2 on fast breaks.
Georgia's Kenny Gaines, coming off career-high 27 points at South Carolina, finished with 10 points.
The Tigers dropped to 2-6 in SEC road games. They finished 2-7 on the road in their first SEC season last year.
Missouri's defense has really struggled in the last two games. Georgia, which began the night with the SEC's worst field-goal shooting percentage in league games, shot 52.1 percent from the field.
The Tigers let Alabama shoot 54.7 percent last weekend -- the Crimson Tide's highest percentage against a Division I team in five years.