Duncan made the first five points of the extra frame and the Spurs went on to outscore the Grizzlies 18-7 in the overtime after trailing by 18 points in the first half. The veteran forward also hauled down 10 rebounds.
Tony Parker added 26 points and 11-of-22 shooting from the field for top-seeded San Antonio, which took a second straight game in overtime.
The Spurs' Manu Ginobili supplied 19 points, seven rebounds and five assists, and Tiago Splitter netted six of his 11 points in overtime.
Mike Conley scored 20 points for Memphis, which managed to hit just 39.2 percent of its shots from the field.
The Spurs will attempt to sweep the series in Monday's Game 4.