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CHICAGO -- The way Ron Gardenhire sees it, the Minnesota Twins really don't own the White Sox. That's probably news to Chicago.

Anthony Swarzak dominated over six innings, and the Minnesota Twins got back to beating the Chicago White Sox with a 6-3 victory on Sunday.

The Twins head into the All-Star break with nine wins in 12 games after taking three of four in this series. They shook off a 4-3 loss on Saturday that snapped a nine-game winning streak against Chicago dating to last season, getting an RBI single by Drew Butera and run-scoring double by Jason Repko in the fourth before tacking on three more while knocking out Jake Peavy (4-3) in the fifth.

That was more than enough for Swarzak (2-2), who simply made it look easy while filling in for the injured Scott Baker (strained right flexor muscle). The 25-year-old right-hander allowed just four hits and one run while striking out five and walking two as Minnesota beat Chicago for the 29th time in 36 games.

"You call it dominance," Gardenhire said. "But it's like a lot of one-run games, a lot of one- and two-run games, anything can happen at the end. We're finding a way right now to get it done. That's a team that can score so many runs. ... They can throw a touchdown in a heartbeat. You sit there as nervous as can be."

In the end, more often than not, he's exhaling, though.

Peavy ran into trouble with two outs in the fourth when he walked Rene Tosoni and Tsuyoshi Nishioka before Butera and Repko came through, making it 2-0, and he didn't make it out of the fifth as the White Sox lost for the fifth time in six games.

Alexi Casilla and Joe Mauer started that rally with singles, putting runners on first and third before Michael Cuddyer made it a three-run game with a sacrifice fly to deep left. Mauer moved up on that play and advanced to third on a single by Danny Valencia before Tosoni -- who had two hits but is headed to Triple-A -- drove him in with a single past a diving shortstop Alexei Ramirez, knocking Peavy out of the game.

Will Ohman came in and the lead hit 5-0 on an odd inning-ending double play, when Valencia scored on Nishioka's grounder to second and Tosoni got caught off the base.

It was another rough outing for Peavy, whose ERA is now at 5.27 after he allowed five runs and 10 hits. He walked two and struck out two, giving him 1,501 in his career. He hasn't performed the way he anticipated after undergoing surgery to repair a torn latissimus dorsi muscle under his right shoulder last summer and plans to see his physical therapist in Alabama during the break.

"No ifs, ands or buts about it," he said. "My body has not bounced back. It's not hard to see. I'm throwing 85-90 mph with my fastball. Breaking balls aren't sharp, can't even strike anybody out with it. Certainly not characteristic, not what I've had in times past, even when I came back here. But I'm looking forward."

Pitching coach Don Cooper had said before the game that Peavy would probably start the final game of the Detroit series, but that plan might be altered.

"I'm 100 percent positive I'm going to come in on the back end of this rotation to give me a week to 10 days off to try to regroup and get ready to start whatever," Peavy said. "Every 5-6 days. I look forward to just getting a little bit of my feet under me."

The Twins, meanwhile, couldn't have asked for more from Swarzak, who was making his fourth start of the season. He allowed a leadoff single to Juan Pierre in the first and didn't allow another hit until Alex Rios singled with one out in the fifth.

The White Sox finally scored with two out in the sixth when Paul Konerko singled in Pierre, and got two more in the seventh when A.J. Pierzynski and Gordon Beckham drove RBI doubles after Rios led off with a walk against Alex Burnett.

After Mark Teahen struck out, Glen Perkins came in and retired Pierre and Ramirez before working a scoreless eighth. Matt Capps then worked the ninth for his 15th save in 21 chances.

Swarzak, however, stole the spotlight before tiring in the sixth.

"They have such a good lineup out there, I was getting ready to throw a fifth pitch out there in the sixth innings," he said. "I tried to throw all four pitchers at them. I said 'off day coming up tomorrow, with the break and everything, just try to keep it quick, keep a good pace and throw the ball over.' And that's what I did today."

Game notes

Gardenhire said after the game that the Twins are activating OF Delmon Young (ankle) from the 15-day disabled list, recalling infielder Trevor Plouffe from Triple-A Rochester and sending C Rene Rivera and OF Tosoni to the minor-league club. ... DH Jim Thome was out of the lineup because of a sprained left big toe after striking out in all four at-bats on Saturday. ... The White Sox placed C Ramon Castro on the 15-day disabled list because of a broken right hand and index finger and recalled catcher Tyler Flowers from Triple-A Charlotte before the game. ... John Danks, out with a strained right oblique muscle, threw three innings Sunday in a rehab start for Triple-A Charlotte and gave up two runs on a pair of homers. Cooper said he'll make another rehab start before rejoining the rotation.