Daniel Palka and Yolmer Sanchez also connected for the White Sox, who had lost nine of 10. Carlos Rodon (1-2) pitched a season-high eight innings in his fourth start after missing the beginning of the year while recovering from shoulder surgery last September.
Moncada, who hit .183 over the previous 35 games, gave the White Sox a 3-2 lead with a three-run double in the fifth. Moncada then scored on Avisail Garcia's single, and Jose Abreu added another RBI single.
Palka, Sanchez and Moncada all went deep in the sixth. Moncada belted a three-run drive for his 10th homer, making it 10-2 Chicago.
Oakland set a modern major league record when Mark Canha hit a drive to left on Rodon's first pitch of the fifth, extending the A's streak to 25 straight road games with a homer. Canha also opened the scoring with a sacrifice fly in the second.
A's right-hander Paul Blackburn (1-2) allowed six runs and eight hits in five-plus innings. He permitted only one baserunner through the first four, but got into trouble with two out in the fifth.
Rodon allowed two runs and seven hits. The eight innings matched his career high.