Johnson drew a one-out walk in the 11th off Leyson Septimo (0-2), and Nate Jones then entered only to surrender a single to Edwin Encarnacion that sent Johnson to third.
Cooper followed by lining a solid hit to right field, delivering a victory for the Blue Jays in the opener of a four-game series.
Steve Delabar (3-1) earned the victory with two innings of shutout relief, during which he fanned six Chicago batters.
Four of the strikeouts came in the 10th inning, when one of Delabar's victims -- Tyler Flowers -- reached base on a third-strike passed ball. It was the first time in club history a Toronto pitcher has recorded four whiffs in an inning.
Moises Sierra went 2-for-3 with a homer for Toronto.
Adam Dunn forced extra innings with a ninth-inning solo homer, his second of the game and 33rd of the year for the American League Central-leading Chisox.