Anderson homered leading off the first for the second straight night and finished with four hits and three RBIs. The reigning AL batting champion raised his average to .379 with his second four-hit game this season - the other was against Detroit last week.
''I'm just seeing the ball real well, just being aggressive in my spot,'' Anderson said. ''Definitely a lot of fun. When you have a lot of fun, the rest takes care of itself.''
Anderson had a pair of singles and a double along with his homer.
''I was definitely trying to get the triple,'' he said, ''but we play again tomorrow.''
Jose Abreu had three hits and three RBIs, and Dylan Cease pitched into the seventh inning to win his fourth straight start as Chicago handed Detroit its seventh loss in a row.
White Sox center fielder Luis Robert was pulled in the eighth inning with a sore right hand. X-rays were negative, and the team says he is day to day.
''All indications are it's nothing of real concern,'' manager Rick Renteria said.
Anderson greeted the left-handed Skubal with a shot to left-center on a 1-1 pitch. He hit two of Chicago's six longballs in a 7-2 win Monday night and has five homers this season.
''Home run, you kind of tip your cap on that one,'' Skubal said. ''It was a good pitch.
''Then another two hits right after that. Just trying to settle down right after that and make some good pitches and quality pitches to get out of that inning, which I did.''
Cease (4-1) allowed two runs and five hits in 6 1/3 innings to improve to 5-0 in five career starts against Detroit. Yoan Moncada had two RBIs.
Cease said he had an effective slider with ''good bite and good shape'' to play off his fastball. But the righty enjoyed seeing his team churn out runs even more.
''I think that's the offense everyone envisioned with all the signings and call-ups we had this year,'' Cease said. ''It's been a blast to watch.''
Jonathan Schoop hit a solo shot off Cease and added a single.
Skubal (0-1) lasted just two innings, allowing four runs and seven hits. The 23-year-old ranks among the top left-handed prospects in baseball.
He and right-hander Casey Mize, the No 1 pick overall in the 2018 draft, were called up to fill a gap in the Tigers injury-depleted rotation. Mize is slated to make his major league debut Wednesday. Neither had pitched above Double-A prior to this season.
After Anderson put Chicago ahead, Schoop lined a homer to almost the same spot to start the second.
The White Sox jumped back ahead 4-1 in the bottom half, with Anderson's double, Moncada's sac fly and Abreu's single driving in the runs. That ended Skubal's night
''I wish I could get a couple of pitches back, but it's part of the game,'' Skubal said.
Abreu drove in two runs with a double off Daniel Norris in the fourth to make it 6-2.
The Tigers scored twice in the eighth. The White Sox replied with four in the bottom half with Anderson's fourth hit, a single, plating Chicago's ninth run.
Anderson is the fifth White Sox player to hit leadoff home runs in back-to-back games. His eight career leadoff homers are tied for fourth on the franchise list.
ANOTHER ROUGH DEBUT
White Sox OF Luis Gonzalez made his major league debut in center as a substitute for Robert in the eighth and dropped JaCoby Jones fly, allowing Detroit's fourth run to score. Gonzalez was hit by a pitch in his first at-bat in the bottom half.
Tigers: OF Harold Castro left in the fifth inning with left hamstring tightness and will have an MRI... INF Niko Goodrum was back in the lineup at shortstop after leaving Sunday's game with right back tightness and sitting out Monday.
White Sox: C Yasmani Grandal is day to day after leaving Monday's night's game with back tightness.
UP NEXT: Mize (0-0, 0.00) takes the mound Wednesday. The White Sox will counter with RHP and highly-rated prospect Dane Dunning, who also makes his major league debut.
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