Gabriel Landeskog scored two goals in regulation and then netted the winner in the shootout to lead the Avalanche to a 5-4 victory on Tuesday night.
With the shootout tied at 1, Landeskog beat goalie Pekka Rinne with a wrist shot from the right side.
Josi scored the game's first goal at 3:33 of the opening period.
After Holden overskated the puck in the neutral zone, Craig Smith carried it into the Avalanche end on the right side and missed wide with a shot from the circle. The rebound came to Josi at the left point, and the Swiss Olympian beat goalie Semyon Varlamov with a slap shot.
"I think we started pretty slow," Varlamov said. "The Predators had so many chances to score. We played better in the second and third periods."
Josi struck again at 9:49 of the first.
Inside the Colorado zone, rookie center Calle Jarnkrok won a faceoff back to Josi. From the center of the blue line, he beat Varlamov with a wrist shot, aided by a screen from Gabiel Bourque, who was stationed just in front of the net.
Jarnkrok has posted an assist in his first three NHL games. He also scored Nashville's lone goal in the shootout.
Landeskog cut the Nashville lead in half with 54 seconds remaining in the first with a wrist shot from the low slot that beat Rinne on the stick side.
Until then, Nashville had controlled the first period. The Predators outshot Colorado 13-3 in the opening frame. Nashville finished with a 33-28 shot advantage.
"It could have been easily three or four nothing for them after the first period," Colorado coach Patrick Roy said. "Sometimes it's nice to win a game when you are not playing well. That was the case tonight."
Colorado tied it at 8:41 when an unchecked Landeskog beat Rinne from the high slot for his second goal of the game. The Avalanche captain set a career high with 23 goals.
Nashville regained the lead at 14:43 of the second when Cullen beat Varlamov from in tight with a wrist shot high to the short side.
"It felt like one of those games where you deserve to win it by the way you played, but you need to give a lot of credit to their team," Cullen said. "They have a lot of dynamic players that can just go out and scored a goal right away. It's unfortunate the way that it ended up, but for a lot of the game, we felt good about things."
After McLeod tied it at 12:25 of the third, Cullen and Holden traded goals 44 seconds apart late to force overtime.
"On the fourth goal, we carried it through the neutral zone instead of making a real firm play, we just sort of threw it in the middle and they came back and transitioned and threw a puck at the net there that got by Pekka," Nashville coach Barry Trotz said. "We didn't lock it down. We didn't make strong plays when we had those leads."
Nashville D Seth Jones missed the game with concussion-like symptoms. The rookie was the only Predators player to be in the team's first 72 games this season. ... Bourque has assists in three straight games. ... Nashville is 19-0-4 when leading after two periods this season... The Avalanche had no power plays for the second time this season.