Eric Robinson and Oliver Bjorkstrand each had a goal and an assist and Seth Jones had three assists for the Blue Jackets (18-26-12). Matiss Kivlenieks made 33 saves to win his first game since his NHL debut Jan. 19, 2020.
Columbus went 3-1-3 in its final seven games of the season.
"I'm glad they're able to leave the season, a horrible season, with a win," Blue Jackets coach John Tortorella said. "For the most part through the time that we were eliminated they played the right way, tried to keep the integrity and the dignity of the hockey for us.
"It was an overtime win, which is exciting for them. I'm proud of them. There's a lot of work to be done but I'm glad we ended at least winning a game."
Jakub Vrana had a goal and an assist and Dennis Cholowski had two assists for the Red Wings (19-27-10). Calvin Pickard made 31 saves in his first start since March 30.
Detroit was 3-2-3 in its final eight games.
"The arc of this organization is starting to head in the right direction and last year was a bad year, for sure," Red Wings coach Jeff Blashill said. "This was totally different. We're a much, much improved hockey team from a season ago, both in how we play the game, in our habits, in our structure.
"With a number, especially down the stretch, of young players that are really being thrust into significant roles, to me, it's not even comparable [to last season]."
The Red Wings and Blue Jackets each finished with 48 points and a .429 points percentage in the eight-team Discover Central Division, but the Red Wings were seventh based on more regulation wins (19-18).
"Down the stretch there I was pretty proud how our team competed," Red Wings defenseman Danny DeKeyser said. "I think a lot of our younger players took strides this year. They're meant to be here. They're playing well. It's good to see that for the future. It gives you a positive outlook on things."
Jack Roslovic gave Columbus a 4-3 lead at 14:45 of the third period before Valtteri Filppula tied it 4-4 at 16:18 off a scramble at the crease.