Roberto Pérez broke the scoreless tie with an RBI infield hit against Sox reliever Garrett Crochet. Pinch hitter Amed Rosario followed with an RBI double.
The Sox went down in order against James Karinchak in the bottom of the 10th. Jake Lamb flied out to the warning track in right for the final out.
Giolito allowed three hits in seven-plus scoreless innings for the Sox. He struck out eight and walked three, exiting after walking Pérez to lead off the eighth with his 107th pitch.
Giolito struck out at least one batter in each of the first seven innings.