The Spurs took a six-point lead with 18 seconds remaining on a free throw by Danny Green on their way to a 15th straight home victory over Atlanta.
Dahntay Jones then hit a jumper to bring the Hawks within four and, after a San Antonio turnover, Jones was fouled trying to make a 3-point shot.
Jones went to the line with a second remaining and hit his first two free throws. He missed the third intentionally in hopes a teammate could grab the rebound, but Kawhi Leonard claimed it for the Spurs to clinch the victory.
Duncan added 14 rebounds and four blocks while Leonard had 23 points. San Antonio moved a half game in front Oklahoma City for the best record in the Western Conference.