Joe Mauer, right, celebrated with Tsuyoshi Nishioka after scoring on a single by Rene Tosoni during the fifth inning Sunday. (Stan Schmidt — Associated Press/The Minnesota Star Tribune)

CHICAGO - Twins manager Ron Gardenhire was asked Sunday to assess his team's performance, now that the All-Star break has arrived. Gardenhire began coughing before he could say anything, then chuckled at the timing.

Health has been an issue, as well as some sickly play earlier in the season. But they have played well enough -- going 24-11 over their past 35 games -- to surge to the fringe of the AL Central division race.

"Given what happened this year with how much time guys missed, key guys, it's pretty remarkable where we are at," Twins catcher Joe Mauer said. "As bad as things have seemed, we have put ourselves within reach."

Their 6-3 victory over the White Sox on Sunday enabled them to take three of four games from Chicago and creep within 6 1/2 games of division leader Detroit. It was a reflection of how the Twins have won over the past month -- getting what they can from fill-ins for injured players.

Righthander Anthony Swarzak, subbing for injured starter Scott Baker, held the White Sox to one run over six innings. Of the first 14 batters he faced, only two reached base. He had a couple of tough innings, but he used all his pitches while improving to 2-2. Paul Konerko got him for an RBI single in the sixth, but Swarzak struck out Carlos Quentin with two on to end the inning.

"That's such a good lineup over there, I was getting ready to invent a pitch out there in that sixth inning," Swarzak said.

Despite the hot streak, the Twins still are seven games under .500. It's a vivid reminder of how bad things were during the first two months of the season.

"We are playing some really good baseball this past month," Gardenhire said, "and we are fighting our way back to .500."

Gardenhire and his players do have reasons to be optimistic when play resumes.

The middle infield looked as if it was going to be a season-long disaster. Alexi Casilla played horribly at shortstop, and second baseman Tsuyoshi Nishioka missed 59 games because of a broken left leg. Casilla was moved to second. Nishioka switched to short when he recovered, and the duo have begun to click.

The offense entered Sunday ranked 25th in runs scored, but the Twins are averaging 5.6 runs over the past 12 games, partially because players such as Ben Revere, Luke Hughes, Rene Tosoni and Casilla have chipped in.

And the bullpen has been remodeled, with Joe Nathan losing his closer's role before landing on the disabled list, Matt Capps taking over as closer and Glen Perkins emerging as a late-inning factor. Capps has struggled of late, but Gardenhire showed faith in him Friday and Sunday by letting him close out victories.

With Nathan looking sharper after his return from the DL, the Twins have three solid late-inning options.

The rest of the bullpen remains an issue, with Gardenhire mentioning Sunday that lefthander Chuck James, currently at Class AAA Rochester, could be called up. And Twins scouts have been monitoring relievers outside the organization.

They continue to hope for the day when they are relatively healthy. Delmon Young will join the team following the break. Denard Span could be back in two weeks, with Jason Kubel sometime after that. That's an entire starting outfield.

Down the road, Justin Morneau could return from neck surgery in August.

The Twins have had to play red-hot baseball to make up for starting 17-37. Will they get healthy, stay hot and jump into the heat of the division race by the final month of the season?

"It's easier to chase than it is to stay on top," Twins outfielder Michael Cuddyer said. "Whether we like being in that role or not I can't say because it's never good being in fourth place."

La Velle E. Neal III • lneal@startribune.com