Stalberg's eighth and ninth goals of the season broke open the contest, handing the Blackhawks a 3-0 lead.
Corey Crawford made 37 stops to record the victory.
The Blue Jackets, who lost their eighth in 10 contests, got their only goal from James Wisniewski.
Steve Mason yielded four goals on 22 shots for Columbus.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Aaron Carter is still in love with Hilary Duff