Hronek tied it 3-3 when he used Drew O'Connor as a screen and beat Juuse Saros on a wrist shot from the right face-off circle with Nikita Tolopilo pulled for the extra attacker.
Marco Rossi had a goal and two assists, and Brock Boeser and Hronek each had a goal and assist for the Canucks (20-37-8), who have two wins in their past 11 games (2-6-3) and four in the past 27 (4-18-5). Tolopilo made 16 saves.
Jake DeBrusk scored in the shootout and Tolopilo stopped Ryan O'Reilly, Filip Forsberg and Steven Stamkos to clinch the win.
Tyson Jost scored twice for his first multigoal game in almost five years for the Predators (29-27-9), who are 2-3-2 in their past seven games. Saros made 24 saves.
Boeser gave the Canucks a 1-0 lead at 8:40 of the first period. Hronek's point shot was tipped by Rossi in the slot and caught a piece of Boeser's stick at the net front before going in past Saros.
Jost tied it 1-1 at 18:28, chipping the rebound off Justin Barron's backhander over a sprawling Tolopilo from in tight.
Jost put Nashville up 2-1 with his second goal at 5:20 of the second period. Roman Josi took a slap shot from the point, and the puck bounced off the glass behind the net and rolled in front, where Jost lifted it over Tolopilo's right shoulder.
Jost had previously scored at least twice in a game five times in his career, most recently on May 13, 2021, as a member of the Colorado Avalanche. He has three goals in his past two games after scoring three in the previous 50 games.
Matthew Wood made it 3-1 at 14:06. He threw the puck toward the crease from the goal line to the right of the net and it deflected in off Canucks forward Liam Ohgren's skate.
Rossi cut the deficit to 3-2 at 15:55, scoring on the rebound of Ohgren's shot from the net front.
