MINNEAPOLIS (AP) — The Chicago White Sox haven’t conceded the division race, during yet another mercurial season for temperamental manager Ozzie Guillen’s team.

Jake Peavy sure pitched like he isn’t giving up.

Peavy won for the first time in more than six weeks with eight shutout innings, and the White Sox beat the Twins 7-0 Sunday to sweep a three-game series in Minnesota for the first time in more than seven years.

“We’re going to grind things out,” Peavy said. “That’s a testament to these guys who have not mailed it in. After that last stretch, we could’ve come in here, our heads hung low, and mailed it in on a team that has had our number.”

Brent Lillibridge, Paul Konerko, Alexei Ramirez and Alex Rios each hit home runs for the White Sox, who had lost 29 of 36 games to their nemesis until winning three straight at Target Field this weekend. The backsliding Twins have lost seven of their last eight games overall.

That’s what the White Sox looked like at the beginning of the weekend, after losing six in a row after finally pulling back to the .500 mark. Those defeats did come against the top two teams in the American League, the Boston Red Sox and New York Yankees, though, and with 50 games left in the underwhelming AL Central the White Sox aren’t at all out of it. They’re still in third place, but they climbed to 5 1/2 games behind the Detroit Tigers.

“It’s all about confidence,” Guillen said. “They’ve got their confidence back, and they know they can win.”

Peavy (5-5) last picked up a victory by pitching four shutout innings in his first career relief appearance, June 25 against the Washington Nationals, three days after he beat the crosstown rival Cubs with a so-so start. The 2007 National League Cy Young Award winner hasn’t been the same since coming to Chicago in a trade with the Padres, but the 30-year-old right-hander was in a rhythm that recalled his old San Diego form.

“Once he got the lead, he just said, `I’m getting ahead of every hitter,”‘ said Minnesota’s Michael Cuddyer, marveling at the way Peavy used and spotted his cut fastball. Cuddyer added: “They stuck it to us, all three games.”

The power surge from Peavy’s teammates sure helped. The White Sox were averaging barely more than four runs per game when he’s pitched, but little Lillibridge went deep for the second straight day in the second inning and Konerko followed suit in the fourth — both shots to the same section in the second deck above left field off Brian Duensing (8-10).

The White Sox hit only 18 home runs in July, their fewest in that month since 1992.

“You have to tip your hat to them. They hit the ball well,” said Duensing, lamenting that he left many of his sinkers up in the strike zone.

Peavy, whose start was pushed back by one day, will probably have to play a big part for the White Sox to catch the Tigers. He gave up only three hits, all in the first three innings, and Joe Mauer’s double was the only way the Twins got a runner to second base.

Peavy struck out six without a walk, hitting one batter.

“We’re just aggressive. I can’t stand putting people on base,” Peavy said. “My stuff was OK, but it makes all the difference in the world when you get a two or three-run lead early.”

Duensing lasted 6 1-3 innings, but he gave up nine hits — Rios had an RBI double to cap a stretch of three straight extra-base hits that started the fourth — and five runs while walking one and striking out four.

One of those runs was unearned, due to a throwing error by Duensing in that rough fourth inning. He stopped a soft comebacker by Gordon Beckham but sent his throw well wide of first base, blowing an easy out and prompting manager Ron Gardenhire to rub his balding head before letting out a big sigh in the dugout. That’s the kind of mistake the Twins have been making lately, and being swept by the White Sox is a sure sign they’re struggling. They were 7-1 against their primary division rival this year until this weekend.

“We have great players here. We had a good series,” Rios said. “Everybody did a good job. That’s what we need, to get things rolling.”

Notes: The White Sox are the only AL team with a winning record on the road and a losing mark at home. They’re 14-5 in their last 19 road games, including five wins in a row. … Guillen gave struggling DH/1B Adam Dunn a day off against a lefty and said he hopes Dunn can start fresh and finish strong this season. The manager also said he wants the slugger to come to spring training next year in better shape. … LH John Danks pitches Monday in Baltimore, facing RH Jeremy Guthrie of the Orioles. … Danks has a 1.69 ERA since June 6, the lowest in the majors in that span.

(TM and © Copyright 2010 The Associated Press. All Rights Reserved. This material may not be published, broadcast, rewritten or redistributed.)

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