Duncan reached a season high in scoring and his improbable jumper from the free throw line with time running out allowed the Spurs to extend their home record to 32-4.
The Spurs let an eight-point, fourth-quarter lead slip away and trailed by one when the Clippers made the mistake of letting the 24-second clock run out without having gotten off a good shot.
San Antonio had 7 seconds remaining in an attempt to come up with the game-winner and Duncan was left to try to put up a jumper from the right side of the free-throw line while being closely guarded by Willie Green.
Duncan's shot bounced off the rim, hit the backboard and then fell through the net to put the Spurs in front. Green was called for a foul and Duncan converted the free throw for the final margin.
It was the sixth win in seven games for San Antonio, which also got 24 points from Tony Parker.
Blake Griffin scored 18 points to lead the Clippers.