"It was rough first inning," White Sox manager Ozzie Guillen said. "I thought it was going to be a little bit worse than that."
Kubel and Hughes each hit a two-run shot off Peavy (6-7), who allowed four consecutive extra-base hits during his rocky beginning. Trevor Plouffe, Joe Mauer and Michael Cuddyer hit three straight doubles to plate the first two runs.
The White Sox fell six games back of the AL Central-leading Tigers, who rallied for a 5-4 victory over Kansas City. Chicago is off Thursday and opens an important three-game series at Detroit on Friday.
There was a stout wind blowing out to left field on a hot afternoon at U.S. Cellular Field and Peavy said Minnesota's ability to get the ball up into the breeze was the key to its strong start.
"They were hitting the ball to left field and the ball was traveling to left field very well," Peavy said. "Obviously I didn't make a lot of good pitches."
The damaging streak of extra-base hits started with Plouffe's one-out double. Mauer drove in Plouffe to give him 500 career RBIs. Cuddyer then doubled in Mauer and scored when Kubel lofted a fly ball to left that got some help from the wind and barely cleared the outstretched glove of a leaping Juan Pierre.
"That ball to Kubel, he put his head down and going up the line, he was upset at himself," Peavy said.
Kubel's third homer in his past four games was his 14th in 39 career games at U.S. Cellular Field.
Scott Diamond (1-2) allowed three runs and three hits in six innings to earn his first major league win. Manager Ron Gardenhire said Diamond may have learned from Peavy's struggles.
"I think Diamond saw that in the first inning and decided he was going to throw the ball down," Gardenhire said.
"That's what Joe is known for is closing games," Diamond said. "That's what always makes the game exciting is ninth-inning drama, but like he always does, he closed the game down and got us a win."
Peavy settled down after the rough start, giving the White Sox a chance to get back into the game. Lillibridge hit a two-run homer in the third and came all the way around from first to score on Konerko's double in the sixth.
"We just battled," Lillibridge said. "I'm sure Ozzie loved the way we played. We played hard and we kept forcing good at-bats and battling to the end."
Peavy allowed six runs and eight hits over five innings, falling to 2-6 with a 5.77 ERA since a relief appearance on June 25. The 2007 NL Cy Young Award winner still isn't 100 percent after having surgery on his right shoulder and said the bullpen outing might have been a mistake.
"If I could go back and take one thing away, I probably set myself back with the relief appearance I had," Peavy said.
Jason Repko added a two-out drive in the seventh for the Twins, who went 7-21 in August for the club's worst-ever record in the month.
Chicago mounted another challenge in the ninth, but Nathan struck out Rios with a runner on first to end the game.
Guillen had Adam Dunn hit for Lillibridge with runners on second and third in the ninth and the much-maligned slugger struck out swinging before Konerko's bloop hit.
"I didn't want the matchup Nathan against Lilli," Guillen said. "The only thing is go out and hit the ball out of the ballpark or walk. Obviously, he struck out but that's the matchup I like the best. "
Tough loss, but now it's on to Detroit for what may be a season-turning series for the White Sox, one way or another.
"I feel like we have to win the series to be really optimistic for the rest of the year," Lillibridge said. "We're ready. We've been playing good baseball."
Chicago OF Carlos Quentin said his sprained left shoulder has improved but he wasn't willing to put a timetable on his possible return. He's been on the disabled list since Aug. 21. ... Twins 1B Justin Morneau (mild concussion symptoms) has to be cleared by the league before he can return. Morneau jammed his shoulder on Sunday and experienced headaches, prompting a concussion test. ... After a day off, the Twins begin a three-game road series against the Angels on Friday with Carl Pavano going against Los Angeles ace Jered Weaver. ... John Danks will start the series opener against Tigers ace Justin Verlander, who is 20-5. Verlander's last loss came against Chicago on July 15.