But the Chicago White Sox are still finding ways to win.
"Any time you can expand a lead I think that those are a lot more meaningful," Dunn said.
The White Sox needed those extra runs as the Royals answered with three in the eighth and one in the ninth.
"Six runs were not enough," Royals manager Ned Yost said. "The eighth inning put a damper on it. We were trying to keep the score at two runs.
"This is a club you go up and down their lineup, you make a mistake and those guys are going to hit it a long way. I guess you call it we caught a break, but it hasn't helped us any missing (Jose) Abreu. He's got a ton of power, too."
Abreu leads the majors with 15 homers, but Dunn and Paul Konerko have hit three-run homers in the first two games of this series.
"Obviously, he's really good," Dunn said of Abreu. "I think Paul and I hopefully can put some runs on the board."
The White Sox out-homered the Royals 5-0 in winning the first two games of this series.
"They've got a lot of power," Royals designated hitter Billy Butler said. "They live and die by the long ball. They hit a lot of home runs and when they don't, they usually lose."
Andre Rienzo (4-0) limited the Royals to two runs and five hits over six innings. He struck out a career-high eight.
"He had some really good stuff today," Dunn said. "He's kept us in pretty much every ball game. ... When he has three or four pitches working like he did tonight, he's tough."
Ronald Belisario gave up a run in the ninth, he stranded the tying run at second base to pick up his first save of the season.
The Royals' record dropped to 5-15 against American League Central opponents.
Beckham has hit safely in 13 of his past 15 games, while Gillaspie increased his average to .347.
Ventura threw three wild pitches in the White Sox three-run fifth, the last one allowing Beckham to score.
White Sox relievers Scott Downs and Frank Francisco combined to retire only one of the six batters they faced in the eighth as the Royals scored three runs. Gordon and Cain had RBIs, while Cain scored on a Francisco wild pitch.
Belisario, who threw 21 pitches in two innings Monday, allowed a two-out single to Gordon, scoring Butler, who had doubled. Jarrod Dyson ran for Gordon and stole second. Belisario struck out Cain on three pitches to end the game.
"It's the same," Belisario said of setting up or closing. "It's the same going in. I do my job. I feel the same."
Sale felt fine after a bullpen session Tuesday and could come off the disabled list this weekend to start against the New York Yankees. ... Royals All-Star C Salvador Perez is sidelined with inflammation in his right thumb. The Royals are hopeful Perez could return Saturday, avoiding a trip to the disabled list. They did recall C Francisco Pena, the son of Yankees bench coach and former Royals manager Tony Pena, from Triple-A Omaha. Pena made his major league debut, catching in the ninth. ... The Royals optioned IF Johnny Giavotella to Omaha. ... Lindstrom, who injured his left ankle trying to field a bunt in the ninth inning, was placed on the 15-day disabled list. The White Sox purchased the contract of RHP Javy Guerra, who had 21 saves for the Dodgers as a rookie in 2011, from Triple-A Charlotte.