The Japanese right-hander also surrendered the first of Nicky Delmonico's two home runs of the night in the fourth and another solo shot to Jose Abreu in the sixth as the White Sox took a 4-2 lead into the ninth.
With runners on the corners, Puig sent a full-count fastball into the gap in left-center for the game-winning two-run double as they Dodgers rallied for their 38th comeback victory of the season and MLB-best 16th sweep of the season.
That season eventual Hall of Fame members included Roy Campanella at catcher, Pee Wee Reese at shortstop, Jackie Robinson in left field, Duke Snider in center field and Red Barber and Vin Scully in the broadcast booth.
In the home debut of Yu Darvish, the Dodgers' pitcher hardly figured into the final outcome on his 31st birthday. He got a force out at home for the second out of the inning but hit Joc Pederson to bring home a run and give the Dodgers the lead. He said that at one time he played for a team that lost 58 games out of the 60 it played, but has never been on one that has won as much as this year's Dodgers have and are expected to continue doing. That quieted the denizens quickly.
After that, the home team blew it open as Austin Barnes and Corey Seager each had a two-RBI single in the inning to keep the White Sox from another upset of one of baseball's best team.
After an off-day on Thursday, the Dodgers will travel to Detroit for a three-game weekend series with the Tigers.
White Sox starter Carlos Rodon made a strong lineup look weak.
Darvish got a no-decision after pitching six innings and leaving with Chicago up 3-2. He walked four and struck out four. Stroman allowed two runs and six hits to beat the Rays for the first time in five career meetings at Rogers Centre. Stanton, who had homered in a team-record six consecutive games, didn't come close to his 45th of the year.
The birthday boy did not enjoy the same run support Wednesday.
Darvish, who is averaging more than a strikeout per inning this season, did not strike out a batter for the first five innings.