The three man-advantage goals were a season high for the Predators.
Martin St. Louis had both goals for the Lightning, losers of three of their last four.
St. Louis scored the game's first goal at 5:26 of the opening period.
Ondrej Palat carried the puck from the left boards into the slot, where he sent a backhand off the left post. St. Louis was parked in front of the Nashville goal to tap home the rebound.
Less than four minutes later, St. Louis struck again.
With the Lightning on a power play, St. Louis' wrist shot from the right faceoff dot hit Nashville goaltender Carter Hutton in the chest, fell to the ice, and trickled across the goal line.
Since scoring a career-best four times in Tampa Bay's Jan. 18 game against San Jose, St. Louis had not scored a goal, a stretch of nine games. He also was pointless in five games for Team Canada in their gold medal-winning performance at the Sochi Olympic Games.
Cullen struck at 12:05 of the second.
Ryan Ellis skated the puck into the right circle, where he faked a shot and found Cullen at the back door for an easy tap-in.
Josi tied the game 1:11 later on another power-play goal when the Swiss Olympian ripped a slap shot from the left faceoff dot over Bishop's left shoulder.
NOTES: Palat has points in three-straight games. . Cullen had not scored a goal since Nov. 27, breaking a stretch of 29 games without a goal. . Nashville is 7-21-5 when its opponents score first.