Reynaldo Lopez had just walked in a run when Shin-Soo Choo came up with a chance to give Texas a lead while extending the longest on-base streak in the majors this season.
"You have to be confident in yourself every time, 100 percent," Lopez said through an interpreter. "Have confidence in your pitches, and execute your pitches and execute a plan. I think that was the key for me that inning."
Avisail Garcia had four hits and Davidson and Jose Abreu drove in three runs apiece to help the White Sox avoid a sweep to start a 10-game road trip, their longest of the season. Davidson had his second straight three-hit game.
Choo extended the on-base streak to 42 games with an eighth-inning walk after going 0 for 4 against Lopez, including the strikeout looking with the bases loaded in a 2-2 tie to end the fourth.
Leading off as the designated hitter after missing a game with a strained right quadriceps that the club said will limit him defensively for a few weeks, Choo's hitting streak ended at 12 games. The on-base streak is the third-longest streak in club history, and longest since 1995.
Lopez (4-5) made it through 6 1/3 innings in the boiling Texas heat — a first-pitch temperature of 97 degrees that climbed past 100 before the last of Lopez's career-high 114 pitches. The right-hander allowed seven hits and two runs with six strikeouts.
"I didn't feel tired in any moment of the game," Lopez said. "But right now, yes. Especially because today's game, a day game with the heat, and I feel really tired."
One-time Texas closer Joakim Soria pitched around two singles in the ninth and had three strikeouts in a five-out save. It was his 12th save of the season in his 14th consecutive scoreless appearance. It was also his first five-out save since 2009.
Five straight Chicago hitters reached with one out in the five-run fifth against Texas ace Cole Hamels, starting with ninth-hitting Adam Engel on a wild pitch after a strikeout.
Tim Anderson doubled, Garcia hit a run-scoring single and Abreu had an infield single before Davidson's double off the bottom of the 14-foot wall in left field.
Hamels (4-7) gave up a season-high seven earned runs on nine hits, which tied the season high from his previous start. The left-hander matched a season low with five innings amid talk of a deadline trade with the Rangers in last place in the AL West.
"The heat's the heat," said Hamels, whose ERA in home starts went up to 5.83 while that number in day games ballooned to 7.53. "It doesn't really have any sort of effect on locating a pitch. I think it's a lot of the times you just get into a rhythm or you get outside of your rhythm and you just don't slow it down."