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KANSAS CITY, Mo. -- Alcides Escobar has been dutifully going to the gym every week, and not just going through the motions. The Royals shortstop has been putting in some work, and truly believes that he's becoming a stronger all-around player.

He made a believer out of a lot of people Saturday night.

Escobar hit a pair of homers off White Sox All-Star Jake Peavy, the second a go-ahead shot with two outs in the seventh inning, sending Kansas City to a morale-boosting 6-3 victory.

"I'm working out in the gym and I'm feeling very comfortable," said Escobar, who came into the game with two homers all season and 11 in his career. "I'm the same guy, though, just trying to hit the ball in the middle."

He certainly accomplished that -- twice.

The 25-year-old Escobar connected on a two-run shot off Peavy (7-6) in the third inning, and after Adam Dunn and Dayan Viciedo helped Chicago knot the game at 3-3 in the sixth, the wiry young shortstop added a solo shot in the seventh for the first multihomer game of his career.

"He's making himself look like an All-Star-caliber-type player," Royals manager Ned Yost said. "We talked about it a bunch, but we knew he was a real athletic player, and once he grew into his body, he would be a real good big league hitter."

Jeff Francoeur and Lorenzo Cain added insurance runs in the eighth, and Greg Holland (3-2) won in relief. Jonathan Broxton rebounded from a blown save the previous night to retire the White Sox in order for his 22nd save of the season.

Broxton pumped his fist after finishing the game -- and all the Royals fans that were in the ballpark Friday night probably did the same. Broxton gave up a run in the ninth in the series opener to turn a potential Royals victory into a 14-inning, 5 1/2-hour affair.

The White Sox eventually won that game on Kevin Youkilis' sacrifice fly.

"He's done really, really well on back-to-back days that he pitches," Yost said of Broxton, who was among the five AL players in the All-Star final vote. "The thing that really impresses me about Brox is that no matter the trouble he gets in, he never gets flustered."

The Royals and White Sox are no strangers to close affairs at Kauffman Stadium, having gone to extra innings in seven of their previous 21 meetings.

This one appeared to be headed the same direction most of the night.

The White Sox struck first when Gordon Beckham, who scored the winning run in the series opener, walked in the third inning, and Alejandro De Aza's double gave Chicago a 1-0 lead.

Kansas City answered in the bottom half of the inning when Jarrod Dyson singled to right, stole second and scored on Alex Gordon's single. Escobar followed with his third home run of the year, this one landing in the bullpen in left field and giving Kansas City a 3-1 lead.

"The pitch in the third to Escobar wasn't a bad pitch, but just the wrong pitch," Peavy said. "It was a ball that he got enough of to hit out of the ball park. Obviously, a big blow there."

The White Sox clawed back, though, just as they repeatedly did the previous night.

Dunn's homer leading off the sixth traveled an estimated 451 feet, clearing two different walls in center field -- "I figured he hit it over the scoreboard," Royals starter Luke Hochevar said. Viciedo added a tying RBI triple off reliever Aaron Crow later in the inning.

The White Sox just couldn't get enough offense to counter the Royals' late surge.

Peavy wound up pitching into the eighth inning for Chicago, throwing 120 pitches along the way. He still ended up losing for the fifth time in six starts. Much of the onus for that falls on his offense, which has managed a total of nine runs of support over that span.

"He was good. Escobar got him a couple of times, those are the ones that got him," White Sox manager Robin Ventura said. "Other than that, he was in it. That's the way it goes."

Game notes

Cain came in as a pinch hitter after starting Friday night. He missed nearly half the season with a multitude of injuries. ... White Sox manager Robin Ventura said Dylan Axelrod and Phil Humber will start against Boston, though he hasn't decided on the order for Tuesday and Wednesday. ... LHP Chris Sale will start the series finale Sunday for the White Sox. RHP Luis Mendoza will take the mound for Kansas City.