As a result, the South Siders' slim hopes of playoff grandeur took another hit. Chicago lost for the sixth time in seven contests and fell eight games behind American League Central-leading Detroit -- tied for the farthest out of first it has been all season.
And with the victory, the Twins increased their lead over the White Sox for second place to 5 1/2 games.
With just 10 games remaining in the regular season for the South Siders, who have an elimination number of four, pride likely will be on the line more so than a possible playoff berth. It appears only a matter of days until the team officially is knocked from contention.
"We either hit and we don't pitch, and then we pitch and we don't hit," White Sox third baseman Gordon Beckham said. "It's kind of the way it's been. If we do both of those, we probably don't play good defense. It's been frustrating, but we showed that we're still fighting, and hopefully, we can get some wins here."
Danks was not sharp from the outset, surrendering three runs in the first frame. Twins shortstop Orlando Cabrera rifled an 0-2 Danks offering over the left-field wall for a two-run home run and a 2-0 Twins lead just two batters into the game.
It was the first of three home runs served up by Danks, who now has given up 25 this season, compared to 15 in 2008. He made it through six innings, allowing seven runs on eight hits.
"What are you going to do?" Danks said. "I could sit here and say I threw a bunch of good pitches and I just got hit. That's just not the case. I wasn't ahead in the count enough, and I let these guys beat me. Who knows? We've dug ourselves a hole for sure. I don't know if we can get out of it or not. We're going to try our [hardest]."
Chicago did show some life offensively, producing its first extra-base hit in 20 innings when Alex Rios homered to left field in the bottom of the third inning off Twins starter Jeff Manship -- Rios' second dinger since being acquired from Toronto. Following a Scott Podsednik single in the same inning, Beckham also launched a home run to left field, his 13th of the season, to put the White Sox in front, 4-3.
The South Siders (73-79) battered Manship, scoring four runs off him in 2 1/3 innings. Manship left after surrendering five consecutive hits with one out in the third inning.
But Danks (12-10) could not hold the advantage, allowing one run in the fourth, another in the fifth and two more in the sixth. Michael Cuddyer's solo home run to lead off the sixth broke a 5-all tie, and the Twins never trailed again, as five relievers combined to allow just two runs over the final 6 2/3 innings.
Cuddyer added to his sparkling statistics lifetime against Danks. He now is hitting .533 (16-for-30) with five home runs and 11 RBIs.
"Cuddyer owns me," Danks said. "I think we should just walk him every time he comes up. I'm at a loss for words. He's a [good] ballplayer. For whatever reason, those guys got something on me or I just serve up cookies. Before long, all nine of them will hit .500 against me, so we'll go from there."
Chicago first baseman Paul Konerko added a solo home run in the eighth, his 28th of the season, to account for the final margin.
Minnesota remained 2 1/2 games behind Detroit for first place, showing it still has much to play for by winning its second straight game against the South Siders.
"Today, maybe, was a must-win for them, too," White Sox manager Ozzie Guillen said. "I think they went out there, our chances are very, very thin and they had a very good chance. I know when they come to Chicago, they know they have a good chance to compete against us and stay close to Detroit, and that's what they're doing."