The Wild sent a second-round pick in the 2017 draft to Buffalo, which will retain half of Stewart's salary, a source confirmed to ESPN.com.
Stewart, 27, has 11 goals and 25 points in 61 games for the Sabres this season. He found out about the deal when reached by phone Monday afternoon by TSN.
"It's kind of been an emotional roller coaster," Stewart said on the network's live trade deadline show. "It's hard not to get caught up in the distractions."
The 34-year-old Leopold, a Golden Valley, Minnesota, native, appeared in 25 games for the Blue Jackets this season and provides the Wild with some depth at the position.
The Wild got a leg up on the trade market earlier this season, acquiring goaltender Devan Dubnyk in a move that has been heralded as a season saver. General manager Chuck Fletcher also added depth forward Sean Bergenheim last week.