The Blue Jackets haven't won in Nashville since April, 3, 2006.
Pekka Rinne denied two of Columbus' three shooters in the tiebreaker to earn his fifth consecutive victory and ninth in his 10 decisions.
Nashville's fifth victory in a row snapped Columbus' three-game winning streak. The Predators last two wins have come in shootouts.
"I have always kind of liked shootouts," Rinne said. "For a goalie, you want to be a difference maker."
Rinne is 2-0 in tiebreakers this season and 7-2 in his NHL career.
On Umberger's first goal, Derick Brassard's pass from the right boards ski-ramped into the air off Erat's stick. Standing in the low slot, Umberger took a baseball swing at the puck and knocked it past Rinne on the glove side.
The goal was initially waved off by referee Tom Kowal, but a prolonged video replay overturned that call. It was determined that Umberger didn't make contact with the puck when it was above the height of the crossbar.
The Predators drew even at 12:12 of the first on a power-play goal by Arnott. Columbus regained the lead 15 seconds later when Umberger scored again.
From below the goal line to the left of the Nashville net, Umberger banked in the puck off Rinne's left skate.
"You get a couple early and it gives you a lot of confidence," Umberger said. "I had a few good chances the rest of the game."
Nashville tied it 2-2 at 1:29 of the second period on a short-handed goal by Legwand.
Erat forced a turnover by Blue Jackets defenseman Jakub Voracek inside the Columbus blue line. Erat found Legwand in front of goalie Steve Mason, and Legwand lifted a backhander high to Mason's stick side.
"I just saw Mason coming out of the corner of my eye, and I tried to get it up on him," Legwand said. "He was down earlier, and I got it up over his blocker."
"The short-handed goal really hurt our momentum a lot because we had full control of the hockey game," Columbus coach Ken Hitchcock said. "It was a careless turnover at the blue line."
Erat gave Nashville its first lead two minutes later with a power-play goal when he redirected Shea Weber's slap shot from the left point. The Predators are 10-0 this season when they score a power-play goal.
"The power play was good tonight," Nashville coach Barry Trotz said. "Special teams were a big part of the game. We got two power-play goals and a short-handed goal."
Stralman's power-play goal at 19:02 of the second period made it 3-3.
Mason stopped 45 shots in regulation and overtime. Rinne turned aside 26 Columbus attempts.
"It's frustrating," Mason said. "They obviously know how to play at home, and they did a really good job."
- Columbus C Samuel Pahlsson returned after missing three games with an elbow infection and back spasms.
- Blue Jackets D Fedor Tyutin has six assists in his last three games.
- Nashville has scored at least one power-play goal in four consecutive games. It's the first time the Predators have done that this season.