Oliver Bjorkstrand had two goals and an assist in his 300th NHL game, and Elvis Merzlikins made 30 saves for the Blue Jackets (17-25-12), who missed the playoffs for the first time in five seasons.
"We're trying to have fun," Merzlikins said. "We give everything we have for these last games. Next season will be a fresh start, and I won't ever remember this season."
The Blue Jackets split the two-game set by getting their second win in 13 games. They lost 4-3 in overtime here Monday.
Mikhail Grigorenko put Columbus ahead 3-2 with a redirection at 13:50 of the third period, and Bjorkstrand scored into an empty net with 1 second left for the 4-2 final.
"I'm trying to be more consistent, just trying to play good hockey," Bjorkstrand said. "It'll be a long summer, but we'll remember how we finished on the year."
Jeannot put Nashville ahead 1-0 at 3:21 of the first period on a wraparound, scoring between Merzlikins' pads and the right post.
"We worked hard," Jeannot said. "We had some sustained pressure out there, just some mini breakdowns, and they made us pay for it. … We've got to refocus, put it behind us and get ready for Carolina."
Jack Roslovic tied it 1-1 with a power-play goal at 2:08 of the second period.
"It was a little bit of a back-and-forth affair, which isn't something we like to do as much as we did tonight," Sissons said.