Trailing 4-2 after the Stars scored four goals in the second period, the Predators began their comeback when Patric Hornqvist scored a power-play goal late in the middle frame.
Mike Fisher supplied the equalizer early in the final period, splitting the defense on a rush up the middle and putting a backhander past Cristopher Nilstorp.
Then, 28 seconds into overtime, Josi hammered a shot from the left point that eluded Nilstorp and broke Nashville's two-game skid.
Nilstorp finished with 26 saves.
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