Senior forward Jordan Barnett matched his career best of five 3-pointers while contributing 15 points and eight rebounds for the Tigers (7-2). Senior guard Kassius Robertson scored 17 points and freshman forward Jeremiah Tilmon had 12.
Junior Darrian Ringo scored 14 points for Miami (5-4), which shot just 32.3 percent from the field. Freshman guard Jalen Adaway added 13 for the RedHawks, who dropped to 8-35 all-time against current SEC programs.
The Tigers never trailed in the contest and are already just one win shy of last season's total (8-24) as they adjust to new coach Cuonzo Martin.
Missouri rattled off eight straight points early in the second half to take a 52-30 lead with over 13 minutes to play.
The Tigers were up 60-37 after Barnett's fifth 3-pointer with 8:36 left before Miami awoke and rolled off 11 straight points to pull within 12 with 5:35 remaining. But Robinson answered with a 3-pointer and Puryear scored the next five, including a 3-pointer with 3:27 left to make it 68-48 as Missouri closed it out.
Barnett scored 12 first-half points as the Tigers possessed a 36-24 lead at the break.
Miami hung with Missouri for the opening 15-plus minutes and trailed 20-19 after Adaway's 3-pointer with 4:50 left in the first half.
The Tigers then caught fire from long range as Barnett and Robinson each knocked down two 3-pointers during a 14-0 burst that made it a 15-point margin with 1:54 left in the half.