Josi, who played for SC Bern prior to joining the Predators, also had an assist.
Filip Forsberg and Matt Duchene also scored for Nashville. Kevin Lankinen made 12 saves.
Sven Baertschi, Romain Loeffel and Christopher DiDomenico scored for Bern. Philip Wuthrich made 16 saves.
Baertschi scored to give Bern a 1-0 lead at 10:57 of the first period, but Josi tied the game when he skated down the right side of the Bern zone and beat Wuthrich with a backhander at 17:46.
Loeffel scored at 19:50 of the first, but Forsberg tied it 2-2 with a goal from in front of the net at 6:24 of the second period.
Duchene put the Predators ahead 3-2 at 8:03 of the second when he scored during a delayed penalty.
Josi made it 4-2 at 5:54 of the third period with a shot from the left face-off dot.
DiDomenico closed the scoring at 19:47 of the third.
