With the score tied 3-3, Elarton gave up singles to Nick Swisher and Alexei Ramirez to put runners on the corners. Orlando Cabrera followed with a single that scored DeWayne Wise, who was pinch-running for Swisher, but was thrown out trying for second.
Pierzynski was intentionally walked and Carlos Quentin followed with an RBI groundout. Jermaine Dye also was walked intentionally and Brian Anderson followed with an RBI double for the 6-3 final margin.
Jim Thome hit a solo home run and finished with two RBI as the White Sox won for the 10th time in 12 games.
Chicago starter Javier Vazquez surrendered three runs in six innings but the win went to Boone Logan (2-1), who threw 1 1/3 innings of perfect relief. Bobby Jenks got his 13th save -- the 100th of his career.
Cleveland starter Paul Byrd also took a no-decision, allowing three runs in 6 2/3 frames, with the loss tagged on Elarton.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys