Beckham began the inning with a walk and made it all the way to third on pinch-hitter Dallas McPherson's single to center. McPherson, who was promoted when Mark Teahen went on the disabled list earlier in the day, picked up his first hit since Sept. 10, 2008, for Florida at Philadelphia.
Eppley (1-1) then bounced his first pitch to Alexei Ramirez, and Beckham rushed home to give Chicago a 4-3 lead.
Jesse Crain (1-1) pitched 1 2-3 scoreless innings to get the win and Sergio Santos worked a perfect ninth for his sixth save in six chances.
The White Sox (18-25) were coming off a listless 4-0 loss to Colby Lewis and the Rangers on Monday night and had dropped 10 of their last 11 games at U.S. Cellular Field. They trailed 3-0 in this one before Brent Morel hit a three-run drive with two out in the fifth for his first homer of the season.
Endy Chavez had two hits and reached four times for Texas, which has dropped eight of its last 11 road games. Michael Young singled in the eighth to extend his road hitting streak to a career-best 17 games.
The Rangers, missing injured sluggers Josh Hamilton and Nelson Cruz, employed a patient approach as John Danks struggled to find the strike zone. They walked five times in the first four innings, with each playing a key role in their three runs.
Chavez walked and scored on Ian Kinsler's double-play grounder in the first. Mike Napoli, Yorvit Torrealba and Craig Gentry walked with two out in the second, and shortstop Ramirez misplayed Chavez's bouncer for a run-scoring error. David Murphy started the fourth with a walk and eventually came around to score on Torrealba's sacrifice fly.
Texas entered the game with 124 walks, 11th in the American League.
Danks eventually settled down, working out of a jam in the fifth and retiring seven of his last nine batters. The left-hander matched a career high with six walks in all, but allowed just two earned runs and four hits over 6 1-3 innings.
Danks, who had dropped his previous five starts, remains winless in six decisions one year after he set a career high with 15 victories.
Matt Harrison pitched seven effective innings for Texas, allowing three hits. He struck out six and walked four.
NOTES: Hamilton (broken bone in upper right arm) took early batting practice, then returned to Texas to be re-evaluated by team physician Dr. Keith Meister. If Meister clears him, the AL MVP will play two rehab games at Double-A Frisco starting Wednesday, then three with Triple-A Round Rock. ... McPherson has been hampered by back problems for much of his career. ... Chicago RHPs Gavin Floyd and Phil Humber will switch spots in the rotation this week. Floyd will start Thursday's game against Cleveland, while Humber is slated to go on Friday against the Los Angeles Dodgers.
Jay Cohen can be reached at http://twitter.com/jcohenap