Up by 14 in the fourth, the Spurs failed to score in the last four minutes of regulation and were pushed to overtime by the rallying Grizzlies.
However, Duncan scored his team's first six points in the extra session to put the Spurs back on track for the Game Two win. The series now shifts to Memphis for Game Three on Saturday.
Memphis also trailed in their opening two series, against the Clippers and Thunder, and will hope they can once again win four straight to advance.
They have yet to lose a home game during the postseason.
Despite trailing for most of Game Two, the Grizzlies ended the fourth quarter on a high and put in seven unanswered points.
Tony Allen drew a flagrant foul with 26 seconds left and cut the deficit to 85-83 with two free throws and Mike Conley hit a short, game-tying jumper.
But Memphis seemed to lose momentum in overtime and their hopes disappeared when Jerryd Bayless missed a potential game-tying three-pointer in the closing seconds.
Bayless tied Conley for a team-leading 18 points, and Zach Randolph finished with 15 and 18 rebounds.
The Spurs' Duncan finished with a team-high 17 points and nine rebounds while Tony Parker added 15 points, 18 assists and three steals.
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