Second baseman Jamey Carroll let a tailor-made double-play grounder go off of his glove in the first inning, eventually leading to the game's first run, and shortstop Brian Dozier's high throw in the second pulled first baseman Justin Morneau off of the bag.
The three errors snapped an eight-game stretch in which Minnesota -- which entered with the fifth-fewest in the Majors -- had committed only one.
"Our defense -- which has been playing really good -- just had a little trouble out there," Twins manager Ron Gardenhire said. "I think we gave them three runs out of six, and it's unfortunate, because we've been catching the ball really, really well."
Those mistakes gave White Sox left-hander Chris Sale all the run support he needed, as he allowed only two hits in seven shutout innings to earn his fifth win of the year.
"Part of pitching is finding a groove and rolling with it," Sale said. "I felt early that I got into a groove and [White Sox catcher A.J. Pierzynski] back there calling one of the best games I've seen him call in a while. It was fun to be out there."
Sale (5-2, 2.50 ERA) struck out six and walked just two, while right-hander Jesse Crain and lefty Matt Thornton pitched a scoreless eighth and ninth, respectively.
It was the fifth time the Twins were shut out this season, as they continued their struggles against left-handed pitching. Minnesota now is hitting only .189 against lefties on the road this season.
"If you don't score any runs you're not going to win anyway, and that's due to their pitching staff over there," Gardenhire said. "The young man threw really well today."
Sale's dominance and the Twins' defensive ugliness led to Minnesota's second loss on its eight-game road trip, which concludes Thursday night in Chicago. The Twins are 5-2 thus far, with much of their success stemming from solid starting pitching and stellar defense.
Left-hander Scott Diamond (3-1, 1.78) did his part, turning in his third quality start in four outings, but dropped his first decision.
Diamond allowed five runs (two earned) on nine hits in six innings, with a two-run home run by Rios in the fourth the biggest blemish.
That pitch, a 90-mph fastball that Rios turned on and sent into the left-field bullpen, caught too much of the plate. Diamond, who has shown great control in his four starts with the Twins, said he needs to focus on nibbling around the strike zone rather than continuously pounding it.
"I think I need to continue to mature a little bit in that sense," Diamond said. "I haven't had a problem throwing strikes so far, so I think I need to learn to expand the zone rather than just constrict it to the strike zone. That's just an adjustment I need to continue to make."
Despite getting little help from his defense, the lefty battled and did well to minimize the sixth inning to only two runs after Mastroianni's error. Pierzynski followed the dropped fly with an RBI double and, after Dayan Viciedo grounded out, Diamond intentionally walked Alexei Ramirez to load the bases.
Rios scored on a groundout by Orlando Hudson, but Diamond struck out Alejandro De Aza to end the inning.
"He threw the ball well. Unfortunately, we didn't make some plays behind him," Gardenhire said of Diamond. "We made him throw a few extra pitches, and made him work a little bit harder. We did turn some double plays behind him and got him out of a few jams, but I thought he was good.
"You take away about three of those runs if we would've caught the ball and we'd have been in the ball game."