Kershaw earned his major-league-leading 15th win as he tossed seven shutout innings in the Los Angeles Dodgers' 1-0 victory over the White Sox.
Los Angeles ran its winning streak to 10 games thanks to Cody Bellinger's first-inning RBI single.
Kershaw (15-2) extended his unbeaten streak to 14 starts since May 6, a stretch that includes 11 victories. He allowed seven hits and one walk while striking out seven.
"My fastball command wasn't great the first few innings but got a little better as it went," Kershaw said. "Nine days off is definitely not something I'm used to, so I'm thankful to get out of that unscathed."
Pedro Baez pitched the eighth, and closer Kenley Jansen worked the ninth for his 24th save.
The White Sox (38-53) dropped their fifth in a row and eighth in their past nine. The Dodgers (65-29) took the opener of a two-game interleague series at Guaranteed Rate Field.
However, the result was overshadowed by a blockbuster trade that sent White Sox third baseman Todd Frazier and relievers Tommy Kahnle and David Robertson to the New York Yankees for reliever Tyler Clippard and three minor-leaguers.
The swap was announced after Tuesday's game. Chicago acquired outfielder Blake Rutherford, left-handed pitcher Ian Clarkin and outfielder Tito Polo.
"This deal's a little bit different from the other ones we've done, but it accomplishes the same goal that we set out from the start," White Sox general manager Rick Hahn said. "That is to add as much impact talent to this organization."
Prior to the trade confirmation, Chicago right-hander Miguel Gonzalez (4-9) worked six innings and took the loss, his fourth straight and ninth in his past 10 starts. Gonzalez, the first of five White Sox pitchers, gave up one run on five hits. He walked five and struck out five more.
Los Angeles' Chris Taylor went career-high-tying 4-for-5 with a run, while Bellinger was 2-for-2 with two walks.
Kershaw had to deal with baserunners in five of seven innings. He brushed off a threat in the sixth as he faced White Sox runners at first and third with one out. Tyler Saladino popped out on a bunt attempt, and Kevan Smith grounded to third to end the inning.
"Seven innings scoreless and pitched out a couple of jams and made pitches when he needed to," Dodgers manager Dave Roberts said. "He wasn't synced up tonight, but when he needed to make pitches, as he always does, he made them."
Bellinger's first-inning single to right-center scored Taylor from second for his 62nd RBI of the season. Corey Seager advanced to third on the hit. Gonzalez walked Logan Forsythe to load the bases but escaped further damage when Joc Pederson grounded into an inning-ending double play.
The Dodgers had the bases loaded with two out in the third, including two walks -- one intentional. However, Gonzalez struck out Pederson to end the threat.
Los Angeles had a chance to add on in the eighth, loading the bases with no outs, but Chicago reliever Dan Jennings struck out Yasmani Grandal, and Yasiel Puig grounded into a double play.
For the game, Los Angeles left 11 runners on base and went 1-for-10 with runners in scoring position. The White Sox stranded eight.
Tuesday's meeting was the teams' first in Chicago since the White Sox took two of three between May 20-22, 2011.
NOTES: Dodgers 3B Justin Turner and White Sox OF Avisail Garcia were named winners of their clubs' Heart and Hustle Award, demonstrating the game's values, spirits and traditions. ... Los Angeles is 37-9 (.804) when Turner and rookie 1B Cody Bellinger (26 home runs) are in the lineup. ... Los Angeles sends RHP Kenta Madea (7-4, 4.38 ERA) against Chicago LHP Carlos Rodon (4-2, 4.32 ERA) in Wednesday's series finale. ... White Sox DH Todd Frazier was pulled from the lineup prior to the game as a healthy scratch amid rumors of the impending trade. Yolmer Sanchez replaced him and went 2-for-4. ... Tuesday's home game was Chicago's 41st, the fewest in the major leagues. ... The White Sox are on pace to win 68 games, the fewest since they went 63-99 in 2013. They currently own the American League's worst record and the third worst in the majors.