Maybe the Royals were just waiting for Independence Day to generate some offensive fireworks. Or perhaps they were simply overdue.</p><p> Either way, they marked the holiday Saturday afternoon by overcoming a three-run deficit in a 6-4 victory over the Chicago White Sox at Kauffman Stadium.</p><p> Alberto Callaspo led the comeback with an RBI triple in the fifth inning that provided the Royals with their first run in 14 innings. He then blooped a two-run single that capped a decisive three-run sixth.</p><p> There was more: an eighth-inning blast from Brayan Peña, a Cuban native celebrating his first July Fourth as an American citizen, and airtight relief from a three-man bullpen relay.</p><p> It all enabled the Royals, for one day anyway, to shed some frustration.</p><p> “Once we got things going…,” center fielder Mitch Maier said. “That’s something we haven’t been able to do the last few days. The way all of our pitchers have pitched, they’ve kept us in every game. We just haven’t been able to produce offensively.”</p><p> The six runs marked the Royals’ biggest output in 20 days -- since a 7-1 victory over Cincinnati on June 14. It also snapped a four-game skid in which they scored just three runs.</p><p> “Guys just kept grinding,” manager Trey Hillman said, “and gave good plate appearances.”</p><p> The Royals, 34-46, managed just seven hits but still overcame a three-run deficit for the first time since May 19, when a four-run rally in the ninth inning produced a 6-5 victory over Cleveland.</p><p> “They put a couple good innings together,” Chicago manager Ozzie Guillen said. “They got big hits. They got clutch hitting. Callaspo’s hit, that was big.”</p><p> Royals starter Luke Hochevar allowed four runs and nine hits in six innings, which halted the rotation’s streak at six straight quality starts. It did, though, mark the 11th time in 12 games that a starter worked at least six innings.</p><p> “We swung the bats well,” Hochevar said. “We played great defense, and the bullpen did an outstanding job. We’ve been playing hard. It hasn’t been a matter of that. We just finally got some hits to fall at the right times.”</p><p> Hochevar, 4-3, got the victory when Juan Cruz, John Bale and Joakim Soria retired all nine hitters over the final three innings. Soria got his 11th save in 13 opportunities -- and his third successive one-two-three save.</p><p> “Everyone looked good today,” Soria said. “That’s important to us. When we start feeling like that, we’ll start winning some games.”</p><p> The White Sox, in search of an eighth straight victory, opened the scoring on DeWayne Wise’s two-out RBI single in the second inning. Jermaine Dye made it 3-0 with a two-run homer in the third.</p><p> Dye’s homer was his 19th homer of the year and, if you’re counting, the 220th of his career since the Royals sent him to Oakland in a 2001 trade that returned shortstop Neifi Perez.</p><p> Chicago starter Gavin Floyd didn’t yield a hit until Mike Jacobs opened the fifth inning with a clean single to right field. Callaspo followed with an RBI triple before Maier’s one-out sacrifice fly made it 3-2.</p><p> Hochevar surrendered a one-out homer to rookie Gordon Beckham in the sixth, but Floyd, 6-6, failed to survive the Royals’ half of the inning.</p><p> Willie Bloomquist led off with a single, and Billy Butler walked. José Guillen’s grounder to second resulted in a force at second -- but no double play thanks, in part, to Butler’s nice takeout slide -- before Mark Teahen delivered an RBI single up the middle.</p><p> That finished Floyd.</p><p> Peña batted for Jacobs and moved the runners to second and third on a weak grounder to third against reliever Matt Thornton. Callaspo then served a soft two-run single into right for a 5-4 lead.</p><p> “I told him that I owe him dinner wherever he wants to go,” Peña said. “That’s what a good team does. Everybody picks people up.”</p><p> That hasn’t happened much lately. Callaspo’s single provided the Royals with their first lead in 39 innings dating to Tuesday’s game against the Twins.</p><p> Peña’s moment came two innings later when he whacked a two-out homer into the right-field bullpen against ex-Royals closer Octavio Dotel.</p><p> “I was trying to make good contact and get on base,” Peña said, “but I really hit the ball well. At the beginning, I didn’t think it had a shot. But then I saw the ball carry. I think I got lucky.”</p><p> The Royals will take that, too. All they can get.