Tyler Flowers reached on a throwing error by third baseman Mike Aviles to begin the ninth. Pinch-runner Leury Garcia stole second before Marcus Semien flied out to right that enabled Garcia to advance to third.
Bryan Shaw (1-1) worked the count full to Sierra, who ripped the payoff pitch just past Lonnie Chisenhall at first to give Chicago the win.
Ronald Belisario (3-3) earned the win despite blowing his third save of the season.
The White Sox had not swept the Indians since May 25-27, 2012.
Notable deaths of 2014 [PHOTOS]