Duncan set up the game-winner by dribbling down the left wing, and then finessed a one-handed fallaway jumper over Emeka Okafor at the foul line with 1.4 seconds left.
The Spurs responded after Carl Landry's tip-in with 5.2 seconds left had pulled the Hornets into a 102-102 tie.
Landry missed a chance to win it with a 3-pointer at the buzzer and New Orleans suffered its eighth straight loss.
Duncan amassed 28 points and seven rebounds while Tony Parker handed out a career-high 17 assists to go with 20 points for San Antonio.
The Hornets were led by 26 points from Jarret Jack.
Landry and Trevor Ariza each contributed 18 for New Orleans.
2014: The Year in Music [PHOTOS]