Reynolds' RBI single in the third inning broke a 2-2 tie, putting the Indians ahead to stay, and his bases-loaded shot deep into the left field bleachers two frames later broke the contest open.
It was the fifth long ball of the season for the Cleveland first baseman.
Nick Swisher and Yan Gomes each added a two-run blast as the Tribe clinched their three-game series against the White Sox.
Zach McAllister (1-1) earned his first victory of the season, controlling Chicago on three runs (only one earned) on five hits through 6 1/3 innings. The right-hander didn't issue a walk and struck out six.
Chisox starter Chris Sale (1-1) was tagged for eight runs on eight hits in 4 1/3 frames.
Alejandro De Aza smacked a two-run homer for the White Sox.