Calle Jarnkrok tied it 1-1 at 6:13 of the second period on a wrist shot off a face-off won by Turris.
Palat gave the Lightning a 2-1 lead 31 seconds later from the slot off a pass from Kucherov at 6:44.
"For us to battle back here and get the two points, it was a Western Conference team, so we were not as worried about giving up that extra point," Tampa Bay coach Jon Cooper said. "The boys have deserved this. We've played some pretty good hockey here, and to have kind of the first period we had where we were a little lethargic, and to come back the way we did, I was pretty proud of the guys."
Predators forward Ryan Johansen was given a major penalty and a game misconduct for elbowing Lightning forward Brayden Point at 9:10 of the second period.
"The puck was coming around the corner and I tried to get body positioning to play the puck," Johansen said. "I haven't seen it, but I guess he ran into my elbow there or my elbow clipped him. I'm just happy to see he was OK and he was back right playing the next shift, so that was nice. It was frustrating."