Norris (1-0) allowed two hits and walked one, earning his first victory since Sept. 28, 2017, and Gordon Beckham homered against his former team as the Tigers won two of three in the series.
Shane Greene had a perfect ninth inning to earn his 10th save in 10 tries for Detroit.
Beckham added a towering solo home run in the seventh inning against his former team.
Tigers reliever Buck Farmer retired all four batters he faced, including three on strikeouts, but the White Sox had a two-out, three-run rally off Blaine Hardy and Victor Alcantara in the eighth. Ryan Cordell, Yolmer Sanchez, Leury Garcia and Tim Anderson all singled to pull within 4-3.
White Sox right-hander Reynaldo Lopez (1-3) had eight strikeouts in six innings, but allowed seven hits and two earned runs.
Dixon and John Hicks had two hits apiece for the Tigers.
Chasing a foul ball from his position at first base, Dixon went into the netting along the wall. He also ran into an out after getting his RBI single on his first Detroit at-bat, then threw wild to second base after Norris may have had Tim Anderson picked off first in the fourth inning.
White Sox: Minor league prospect Luis Robert left Saturday's game for High-A Winston Salem with a left hand contusion. He's day to day. . RHP Lucas Giolito said he hopes to miss only one start after hurting his hamstring on Wednesday in Kansas City.
White Sox: On Monday, LHP Manny Banuelos (1-0, 3.48) will make his first start since Sept. 6, 2015 when he was with Atlanta. He'll oppose RHP David Hess (1-3, 5.57) in the opener of a three-game series in Baltimore.