Rodgers' basket put the Jaguars ahead 54-53, and the Tigers couldn't make their final two shot attempts to get back in front.
The win sends Southern back to the NCAA Tournament for the first time since 2013. The Jaguars were ineligible for the last two seasons.
Rodgers finished the game with 11 rebounds and seven points in the win. Southern was led by coach Roman Banks' son Trelun, who scored 19 points, including two big second-half 3-point baskets.
Raeford Worsham had 17 points and 12 rebounds for the Tigers. Paris Collins, who was battling a shoulder injury that has plagued him all season, added 11 points and seven rebounds, but shot just 3-for-13 from the floor. As a team the Tigers shot just 26.3 percent and were 5-for-20 from 3-point range.
Southern will enter the NCAA Tournament with a record of 19-12, but they only played a single game against a team in the top 100 RPI, a 70-56 loss to Grand Canyon early in the year. The Jaguars also have three losses against teams with 300-plus RPIs and two more against teams that have RPIs between 250 and 300. That sounds like a recipe for Dayton if there ever was one.