Columbus goalie Elvis Merzlikins made the save on Dylan Larkin's shot on the first rush of overtime, but the rebound went in off Bertuzzi's leg for his 50th NHL goal.
"I was just trying to go strong to the net and the shot hit me in the leg and went in, but I'll take it," Bertuzzi said. "We played really well for 60 minutes tonight, and I thought we deserved to win this one."
Anthony Mantha and Bobby Ryan scored, Larkin had two assists, and Jonathan Bernier made 22 saves for the Red Wings (2-2-0), who lost 3-2 to the Blue Jackets on Monday.
"It's very significant for us to win two of these first four games," Larkin said. "I really like the way we've played every game with the puck, we've had a lot of confidence, and our goaltending has been great. Starting the season this way has been huge."
Boone Jenner and Alexandre Texier scored for Columbus (1-2-1). Merzlikins made 24 saves.
"We just haven't developed enough in these games; we've been sloppy," Blue Jackets coach John Tortorella said. "There wasn't much offense out there for either team."
Red Wings coach Jeff Blashill split up the usual top line of Bertuzzi, Larkin and Mantha, but they remained on the same power-play unit and Larkin and Bertuzzi were together to start overtime.
"I wasn't trying to send any message; I just wanted to spread out our scorers and see if we could develop some new chemistry," Blashill said. "It evened out the playing time a little bit, and everyone responded with a good game."