The victory tied Commodores coach Kevin Stallings for most wins as Vanderbilt head coach (278) with Roy Skinner (1961-76).
Odom hit five 3-pointers for the game and sank a pair of free throws with 9.8 seconds for an 84-80 lead. Eric McClellan, a transfer from Tulsa, added 18 points in his debut, including two fouls shots for the final points.
Jones hit a jumper and Dai-Jon Parker sank a 3-pointer to push Vanderbilt ahead 78-73. VU's Kyle Fuller then hit a pair of free throws with 45.1 seconds left for the largest lead (80-74) since 8:09 remaining.
Devonta White's two free throws had cut it to 82-80 with 10 seconds remaining for the Panthers, picked as a preseason Sun Belt favorite in several national media outlets. GSU is now 4-28 against Southeastern Conference teams.
Curtis Washington scored seven straight points as GSU built a 46-42 advantage early in the second half, capped by White's pull-up jumper.
McClellan's second 3-pointer put Vanderbilt back ahead 57-54, while Jones blocked a shot on the other end and Shelby Moats' basket stretched it to 59-54. The Commodores took their largest lead with 8:09 left, 64-58, on Odom's third trey and his assist to Moats forced a Panthers timeout.
Georgia State came out hot as four different shooters sank 3-pointers to take a 12-7 lead with just over 4 minutes to play.
Odom hit two 3s for Vanderbilt early, closing it to 12-9. Jones then gave the Commodores a boost off the bench with five straight points including a dunk to launch a 10-2 run for a 20-16 lead.
There were eight lead changes the first half and two times. Harrow hit two free throws, then had a steal and layup to start a 9-2 spurt and put the Panthers ahead 25-22.
But Dai-Jon Parker hit two treys during an 11-2 run that gave Vanderbilt its largest first-half lead at 33-27. Harrow had four points as GSU outscored VU to pull within 35-34 at halftime.
Stallings, in his 15th season at Vanderbilt, is now 14-1 in season-openers.
Mike Krzyzewski says he has good reason to be excited for next season thanks to a combination of Duke's incoming freshmen and a core of experienced upperclassmen.
Georgia guard Juwan Parker has been awarded a medical redshirt, leaving him with two years of eligibility, after missing the 2015-16 season with a partially torn Achilles tendon.
Forward Trevon Bluiett has withdrawn from consideration from the NBA draft and will return to Xavier for his junior season.