The Red Wings' Henrik Zetterberg was whistled for hooking Patrick Kane with 4 seconds remaining in regulation, giving the Hawks a 4-on-3 advantage in the extra frame.
Hossa positioned himself in the left circle and blasted a slap shot past Joey MacDonald, handing Chicago a much-needed victory as it began a pivotal three-game road trip while clinging to the final Western Conference playoff spot.
Jonathan Toews and Brent Seabrook scored and Corey Crawford stopped 33-of-35 shots for Chicago.
Zetterberg and Niklas Lidstrom scored for the Central Division-leading Red Wings, who remained tied with San Jose for the second overall seed in the West.
MacDonald finished with 38 saves.
Boston schools pull out free condoms over wrapping complaints
NBC reportedly holds celebs hostage to Jimmy Fallon's show