Dempster (7-9) left with the lead after working 6 1/3 innings, during which he gave up three runs on five hits with five strikeouts. It was only his second win in 10 starts.
His previous outing had come on Aug. 18, a game against the New York Yankees in which he threw three straight pitches at Alex Rodriguez before hitting him with a fourth. He was suspended five games for that action and reportedly was in danger of losing his spot in the Boston rotation.
Mike Napoli gave Dempster the lead in the third by drawing a bases-loaded walk and Boston scored three times in the fourth, two of the runs coming in on a single by Ortiz.
Chicago cut its deficit to one run through seven innings, but three Boston relievers combined to hold the White Sox to one hit over the final 2 1/3. Koji Uehara struck out two in the ninth for his 15th save.
Hector Santiago (4-8) lasted just 3 2/3 innings to take the loss, giving up the four Boston runs on five hits.
Celebrity Families of 2014 [PHOTOS]