The Boucherville, Que., native finished behind gold-medal winner Victor An of Russia and China's Dajing Wu.
Cournoyer advanced to the finals with second-place results in the quarter-finals and semifinals. He was assured of a medal when China's Wenhao Liang fell during the four-skater final.
The 22-year-old, who was making his Olympic debut, hadn't finished better than sixth during the current World Cup season.
Cournoyer was the only Canadian in the final after Olivier Jean of Lachenaie, Que., was fourth in his quarter-final race to finish 13th overall.
Charles Hamelin of Saint-Julie, Que., was the defending champion but failed to advance out of the heats.
The medal will be a relief to a short-track team that has just two medals in a disappointing overall performance at the Winter Games.