Quintana (9-6) yielded two runs on five hits, walking one and striking out six en route to the win.
Donnie Veal and Nate Jones each recorded an out to work out of an eighth-inning jam and Addison Reed recorded his 39th save as the Chisox won the makeup game of a June 12 rainout.
Marcus Semien clubbed a two-run homer and Avisail Garcia added a solo shot for the White Sox.
Blue Jays starter J.A. Happ (4-7) was touched for three runs on eight hits in a five-inning effort.
J.P. Arencibia homered for Toronto.
Swim Week Miami Beach 2014 [PHOTOS]