Stuart's score came 3:11 into overtime on a long pass from Dan Cleary, who notched a pair of assists in Detroit's win.
The Wings also got goals from Patrick Eaves and Tomas Holmstrom as they clinched the fifth spot in the conference standings. Detroit will face the Phoenix Coyotes in the opening round.
Jimmy Howard posted 27 saves for the Red Wings.
Dave Bolland and Colin Fraser tallied for the Blackhawks, who settled for the No. 2 seed and will face seventh-seeded Nashville in the first playoff round.