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.