Georgia (19-9, 10-6 Southeastern Conference) has won three straight games, including road wins against Alabama and Mississippi, since back-to-back disappointing home losses to Auburn and South Carolina.
"We can't celebrate too much about this win," said Frazier, who had 10 points, five assists and three steals. "We gave away two games at home. We made it tough on ourselves. Our back was against the wall, but we're fighting out of it."
The Bulldogs, 5-3 on the road in conference games, needed to avoid a third straight home loss to one of the league's worst teams. Georgia began the day with a No. 29 ranking in the NCAA RPI and plays host to No. 1 Kentucky on Tuesday night.
Missouri (8-21, 2-14) couldn't manage a second straight win after ending a 13-game losing streak by beating Florida 64-52 on Tuesday night. The Tigers, last in the SEC, trailed by double figures most of the game.
The Wildcats made only 14 of 49 shots from the field (28.6 percent). Missouri had worse shooting in only one other game this season, an 86-37 loss at Kentucky in which it made only 27.1 percent of its shots (16 of 59).
Missouri coach Kim Anderson said his players couldn't match Georgia's intensity.
"To their credit, (Georgia) came out today ready to play and it's frustrating," Anderson said. "In the first half especially, our front line really struggled to score. You can't do that."
Anderson said the Bulldogs "got every loose ball and every rebound that flew off."
A 3-pointer by Djurisic gave Georgia an 18-8 lead. The Bulldogs led 30-18 following a 3 by Kenny Paul Geno, and they kept the lead in double figures the remainder of the game.
"We definitely made an emphasis that we wanted to come out and compete from the tip," said forward Marcus Thornton.
Georgia led 41-23 at halftime and stretched the lead to 28 points at 55-27 on two straight baskets by Cameron Forte -- a follow-up layup and a jam. Following Forte's jam, the Bulldogs kept the lead over 20 points despite going more than six minutes without scoring.
Forte had 10 points for Georgia. Thornton finished with eight points and seven rebounds.