Moore (12-3) scattered five hits over 6 1/3 innings, yielding two walks and striking out six in logging his fourth win in four starts.
Desmond Jennings, Luke Scott and Sam Fuld each drove in a run for Tampa Bay.
The loss was hung on Chisox starter Chris Sale (5-8), who allowed just two earned runs on six hits in seven frames. He walked one and fanned nine.
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