Granlund tied it 2-2 on a rebound for a power-play goal.
"We didn't have our best game in the first period, but we started getting our game together in the second and just kept going in the third," Granlund said. "We keep finding ways to win these games."
Roman Josi had two assists for the Predators (21-18-1), who have won 10 of 12. Juuse Saros made 25 saves.
"It was a tough game to play, and I thought the Red Wings outplayed us in the first period," Nashville coach John Hynes said. "I really liked the way we stayed with it and warmed to the task."
Thomas Greiss made 29 saves for the Red Wings (13-22-6), who have a point in five of seven.
"I thought we had a good start to the game, but we had a lead in the third period, and we found another way to lose," Detroit captain Dylan Larkin said. "We're making progress, but at some point that has to turn into getting results."
Ryan Johansen and Josi scored in the first two rounds of the shootout, and Saros made saves against Larkin and Filip Zadina.
"Juuse is just really good at shootouts; we see that every day in practice," Josi said. "I've taken two and I've scored both of them. I don't work on them much, but we all have our signature move to give ourselves some options."