Mississippi State (2-0) trailed for most of the second half before storming back in the final few minutes. Four Mountaineers (2-1) fouled out of the game, giving the Bulldogs plenty of scoring opportunities at the free-throw line.
With 21 seconds remaining, reserve guard Jalen Steele drove the floor for a transition layup to secure the victory for the Bulldogs.
Augustus finished with 26 points and 10 rebounds, while Johnson added 19 points on 6-of-13 shooting.
Donald Sims scored 40 points for Appalachian State, but had a crucial turnover on the Mountaineers' final possession.
The Mountaineers made 25 of 67 (37.3 percent) shots from the field and turned the ball over 10 times.
The Bulldogs were 25 of 57 (43.9 percent) from the field and 8 of 22 (36.4 percent) from beyond the arc.
Mississippi State led for most of the first period and 43-32 at the half, but Appalachian State closed the gap quickly in the second half. Sims tied the game at 53 on a 3-pointer with 11:43 remaining, then gave the Mountaineers the lead with a layup and free throw on their next possession.
Sims was 11 of 31 from the field, including 7 of 20 from beyond the 3-point arc.
Mississippi State led briefly again around the 8-minute mark. Appalachian State quickly regained the lead and stretched its lead to 73-64 with 2 minutes remaining, but let the game slip away.
Nathan Healy added 13 points and nine rebounds for the Mountaineers.
Renardo Sidney and Dee Bost, two projected starters for the Bulldogs, did not play because of NCAA suspensions.