Rios' double bounced just inside first base to score Alejando De Aza from third. De Aza led off the game with a bloop single, took second on Brent Morel's tapper in front of the plate and advanced to third on what was scored an errant pickoff by Kershaw that shortstop Hanley Ramirez apparently lost sight of in the glare.
Beckham's was a two-out liner past third baseman Luis Cruz that scored Alexi Ramirez, who had singled back through the box. Kershaw struck out three without a walk.
Chicago scored two more off Ronald Belisario in the third inning, loading the bases with no outs and scoring on Dayan Viciedo's two-run single that bounced into left field. Tyler Flowers made the lead 5-0 with a home run leading off the fourth inning against Stephen Fife.
"It's competition. That's what happens down here," said White Sox manager Robin Ventura of the impressive opener. "You don't know exactly what's going to happen at the end with that, but you want them competing and getting after it and earning a spot."
Juan Uribe had the first hit for the Dodgers, a line single, and Andre Ethier the first extra-base hit, a triple off left-hander Leyson Septimo.
Up next for White Sox: With frontline starting pitchers not scheduled to start until Chris Sale takes the mound on March 1, some of the younger hurlers are getting a first-week look. Erik Johnson gets the call Sunday at 2:05 p.m. CT on Gameday Audio in a rematch against the Dodgers, with Zack Greinke on the mound for Los Angeles. Jeff Keppinger is scheduled to make his White Sox debut, starting in the designated hitter's spot.
Up next for Dodgers: Zack Greinke and Hyun-Jin Ryu, the $200 million upgrades to the starting rotation, make their Dodgers exhibition debut Sunday against the White Sox, at noon PT on Gameday Audio. Greinke will start and is expected to go two innings, followed by Ryu, who more likely will pitch only one inning. Although the Dodgers have eight starting pitchers in camp with guaranteed salaries, Greinke and Ryu are already slotted into the rotation behind Opening Day starter Clayton Kershaw. While Greinke is a proven Major League star, curiosity surrounds Ryu, who was the best pitcher in Korea.
Ken Gurnick is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.