The 26-year-old carries a US$1.25-million cap hit through next season.
In 51 games with the Blue Jackets this year, Clitsome had four goals, 14 points and minus-6 rating.
A ninth-round draft pick by Columbus in 2004, Clitsome says he's excited to be heading to Winnipeg.
"It'll be different for me. I've never had that experience throughout my whole career," Clitsome said. "It'll be a learning experience and I'm really looking forward to it.
"I'm excited. They're in a playoff push and they've got a great team. It's going to be exciting times."
Jets coach Claude Noel remembers Clitsome from when he was an assistant coach in Columbus and took over as interim head coach during the 2009-10 season.
"I liked what he brought to the table," Noel said. "I didn't have him for a long time but I liked him. He was a smart player. I thought he shoots the puck real well. He's got good hockey intelligence."
Noel said he wasn't expecting Clitsome to dress for Monday night's game against Edmonton in Winnipeg.