Marson hit a sacrifice fly in the bottom of the second and an RBI single in the fourth.
Travis Hafner had two of Cleveland's six hits -- including a solo homer. Trevor Crowe hit a sacrifice fly, while Matt LaPorta hit an RBI double for the Indians, who won their first game following the dismissal Wednesday of Manager Eric Wedge.
Winner Fausto Carmona (5-12) surrendered a run and five hits in seven innings with two walks and five strikeouts.
Chicago's run scored on a fourth-inning double by Dewayne Wise. Mark Kotsay had two of the six hits for the White Sox, who had won their last two outings.
The losing pitcher was starter Carlos Torres (1-2).
Celebrity Families of 2014 [PHOTOS]