Alejandro De Aza contributed a bases-empty shot as the White Sox avoided a sweep at the hands of the Indians.
Jake Peavy (2-1) surrendered just one run on five hits over seven innings, striking out 11 while improving to 4-0 in his last five starts against Cleveland.
Addison Reed secured Chicago's first road win over year with his fourth save.
Tribe starter Brett Myers (0-2) gave up six hits over five innings, including his eighth homer in 16 1/3 innings to start the season.
Michael Bourn exited the game with a hand injury after going 3-for-4 with a leadoff homer for Cleveland.
Celebrity Couples of 2014 [PHOTOS]