With the game tied 1-1 in the sixth inning, Vizquel led off against Indians starter Justin Masterson (8-7) with a single and two outs later Paul Konerko was intentionally walked, bringing up Adam Dunn.
The Chisox designated hitter lofted a deep fly ball that center fielder Ezequiel Carrera lost in the sun allowed to fall in, scoring both runners for a lead the White Sox wouldn't relinquish.
Edwin Jackson (7-7) lasted six innings, allowing two runs on five hits, to earn the win for the White Sox, who swept a rain-shortened two-game series from the Indians.
Jesse Crain and Chris Sale combined for 1 1/3 scoreless relief innings and Sergio Santos collected his 20th save as Chicago moved to within two games of .500 (49-51).
Masterson was tagged with the loss after being victimized by three Indians errors. He yielded four runs (one earned) on four hits in a seven-inning effort.