MINNEAPOLIS — 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 5½ 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.”