The Buccaneers (2-1) led by 21 points with nine minutes to go before guard Donnell Covington picked up his fourth foul.
The Bulldogs (1-2) immediately put on a full-court press and went on a 21-4 run to cut Charleston Southern's lead to 62-58 with two minutes to go.
After trading baskets, Moore drew an intentional foul as he made a lay-up with 22 seconds to go. He made both free throws and Jackson added two more foul shots after the Buccaneers got the ball back to take a 70-60 lead.
Charleston Southern shot 50 percent (27-of-54) from the field and outrebounded the Bulldogs 38-22.
Covington had 14 points, six rebounds and five assists for Charleston Southern, who is now 3-0 against The Citadel and the College of Charleston under second-year coach Barclay Radebaugh.