Harden (1-1) allowed three runs and four hits after lasting less than five innings due to wildness in three of his first four starts. This time he hit two batters and walked five, giving him 23 walks in his five starts.
The White Sox were unable to capitalize until Carlos Quentin ended a 2 for 37 slide with a three-run homer off Harden in the sixth that reduced the deficit to 5-3.
"I got through six and that's progress," Harden said. "I want to work on keeping my walks down. That's going to be the key. There are a few thing I've been working. Mechanically, I've been trying to find my arm slot. Mentally, I need to be aggressive. Sometimes I try to be too fine."
The Rangers signed Harden as a free agent in December to take over one of the top two spots in their rotation, but Harden's once-dynamic fastball has been less than imposing so far.
"I thought Rich was much better," Rangers manager Ron Washington said. "He was in trouble, but he made the pitches when he had to. It's a start."
Harden was the beneficiary of three double plays.
"It was huge for me tonight," Harden said. "The defense played great. The plan is to get the ball in play. We have solid infielders who can turn those double plays."
"Harden would do everything to give up runs, but he'd make one pitch every inning to get out," White Sox manager Ozzie Guillen said.
Neftali Feliz allowed RBI singles in the ninth to A.J. Pierzynski and Alexei Ramirez before Juan Pierre was out on a close play at first after a grounder to shortstop for Feliz's third save in four chances.
Chicago starter Jake Peavy (0-2) was in trouble early as Hamilton's 430-foot blast got the Rangers rolling in the first. One out later, Peavy walked three straight.
But Peavy got untracked, retiring 16 of the next 18 from the first through the sixth.
"He was all over the place, but then he started throwing more strikes and was more aggressive in the strike zone to give himself a chance," Guillen said. "Very, very ugly beginning, but late in the game we fought back."
Peavy gave up six runs and six hits in 6 1/3 innings with seven strikeouts and five walks.
"Anyone watching the game could see it coming together, but it's frustrating when you can't make a pitch in the first inning to minimize that damage," Peavy said. "Just make one pitch to Arias, to Borbon, and I feel like we had a good chance to win that game. The end result was frustrating."
Guillen didn't have slumping second baseman Gordon Beckham in his starting lineup for the third time in 21 games this season.
Beckham's second major league season is off to a 15-for-73 start with four hits in his last 33 at bats.
Guillen said Beckham is simply trying too hard.
"Like I say every day, you can't come here and think you're carrying this ballclub," Guillen said. "This is about 25 guys, not about one. He's trying to do too much, he's desperate to do good."
Peavy allowed seven runs and seven hits with seven walks over 4 1/3 innings in his previous start, a 10-2 loss to Tampa Bay last Thursday. ... Guillen said Beckham would be back in the lineup for Thursday's series finale against Texas. ... Before the game the Rangers reinstated RHP Tommy Hunter from the 15-day disabled list and assigned him to Triple-A Oklahoma City. Hunter, who's been on the DL since April 4 with a left oblique strain, threw two perfect innings in an extended spring training game in Surprise, Ariz. on Wednesday and will throw in the bullpen in Arizona on Friday before reporting to Oklahoma City. ... Peavy threw 41 pitches in the first as the Rangers sent 10 to the plate.