Nashville forward Steven Stamkos tied the game 2-2 at 19:59 of third period when he buried a wrist shot from the top of the left face-off circle.
Will Smith and Mason Marchment scored for the Sharks (4-3-0), who had lost three of their past four games.
Ryan O'Reilly scored for the Predators (2-2-3), who have lost four of their past five (1-1-3). Juuse Saros made 30 saves.
Tyler Toffoli’s no-look pass from below the goal line to Marchment gave San Jose a 1-0 lead at 5:30 of the first period. Toffoli corralled a loose puck behind the net and fed a pass to Marchment in the slot, where he sent a one-timer past Saros.
Smith made it 2-0 at 4:28 of the third period. Michael Kesselring sent a backhand pass to Smith from the right boards, and Smith skated into the right circle before burying a wrist shot to the blocker side.
O'Reilly cut the deficit to 2-1 with a power-play goal at 9:05. Jonathan Marchessault took the initial one-timer from the left circle, and O'Reilly chipped the rebound in under Nedeljkovic's arm from in front.
Nedeljkovic robbed Predators defenseman Roman Josi with nine seconds remaining in overtime when he lunged across the crease to make a glove save.
