Cal Petersen made 29 saves before stopping Filip Forsberg, Matt Duchene and Roman Josi in the shootout.
"When you do stuff like this, it just goes to show [that] when we inevitably get in these sort of games where we are down or seemingly out of it, we can reach back and know that … we've done it before," Petersen said. "These are huge for our confidence. I think this is what defines a team. I think it's incredible that we were able to do that."
Mikael Granlund had two assists, and Juuse Saros made 32 saves for the Predators (2-2-1), who have lost three straight.
Nashville was penalized nine times in the game, allowing one power-play goal and one at 4-on-4 after a penalty negated their own man-advantage.
"It starts with the penalties," Predators coach John Hynes said. "Offensive-zone penalties, penalties on the power play, penalties at key times. You can't get any momentum when you do that. You're putting their top guys on the ice.
"I thought the third period, our decision-making with the puck didn't set us up to be able to play in the attack. We had to take on a lot of water."
Roy cut it to 3-2 at 13:00 of the third period when Brendan Lemieux's centering pass redirected in off his right shin on a 3-on-2. He then tied it 3-3 at 17:52, beating Saros from the right circle during a 4-on-4.