While not as ugly as Tuesday's 19-2 blowout, the setback was the Rangers' third in a row and snapped their seven-series winning streak. The Rangers are mired in their third three-game losing streak of the season and first since they dropped four straight from May 29-June 2.
"We had our chances, we didn't cash them in," manager Ron Washington said. "We're still  games above .500, we're still in first place. We'll show up tomorrow and play."
Adams (1-3) led off the 10th by walking Alejandro De Aza after getting ahead of him, 0-2. De Aza fouled a pitch off his knee, which delayed the game for several minutes as he shook it off, but he stayed in to draw the walk.
Adams, who had pitched a perfect ninth inning, said he felt like De Aza and the White Sox were "icing the kicker."
"It's always frustrating when they get an extended period of time like that. I understand he fouls it off, but either get out of the game or stay in the game, one or the other," Adams said. "But there's nothing I can do about it. I have to do a better job of staying with it and coming out being focused."
De Aza then stole second before Youkilis knocked out the game-winning hit on the ninth pitch of an at-bat that included six foul balls and a check swing that was called a ball.
Washington said he thought Youkilis went, while Adams said, "It was close. Sometimes those get called, sometimes they don't. A lot just depends on the umpire."
Youkilis, a recent addition to the White Sox, said he was just trying to stay back after nearly getting rung up.
"I was a little jumpy on the slider and out in front of it. I saw it good and the last time I told myself to stay back a little bit as much as I can," Youkilis said. "If there's a fastball, throw the hands in as quick as I can. So that happened, I got the pitch and I was very fortunate to get it over [Rangers shortstop Elvis Andrus'] head."
Texas had chances to take the lead late, putting two runners on with two outs in the sixth and seventh innings, but White Sox right-hander Nate Jones got out of both jams.
The White Sox also had two on with two outs in the seventh before Rangers rookie left-hander Robbie Ross caught Adam Dunn looking to end the inning.
Ross tossed 1 2/3 innings to increase his scoreless streak to 21 innings and lower his ERA to 0.99.
"I know they have pitching, I know we have good pitching and the good pitching showed up," Washington said. "When you pitch, there won't be any more hitting, and that's exactly what happened. We had opportunities, we just didn't cash them in. They had opportunities. It just took 10 innings before someone got a big hit, and Youkilis came up big for them."
The Rangers staked starter Scott Feldman to a three-run lead with a two-out rally in the first inning, and Josh Hamilton added solo home run in the third, his 26th of the season.
Hamilton began the first with a single off White Sox starter Dylan Axelrod, who gave up another single to Adrian Beltre. Michael Young drove in both with a double -- breaking an 0-for-13 drought. Young then scored on Texas' fourth straight hit, a single by Nelson Cruz.
The White Sox answered with four runs in second against Feldman. After Alex Rios' one-out double, Feldman hit A.J. Pierzynski with an 0-2 pitch. Dayan Viciedo and Alexei Ramirez followed with consecutive RBI singles. Pierzynski's run put the White Sox within one after he jarred the ball loose on a collision with catcher Mike Napoli, then Gordon Beckham put the White Sox ahead, 4-3, with a two-run double.
Feldman and Axelrod rebounded after their early struggles. Feldman allowed only the four second-inning runs on six hits in six innings, while Axelrod scattered seven hits in 5 2/3 innings.
"Giving up a four-spot there after we went and put up three wasn't the ideal situation, but I was able to make better pitches after that," Feldman said. "That second inning there I was just kind of leaving the ball up and they were able to get some bat on the ball and find some holes."