The first inning doomed Jeff Francis again on Sunday.
Francis (4-13) has yielded 26 first-inning runs in 25 starts this year.
"As long as I've been pitching, that's been somewhat of an issue," he said. "You try everything you can. I think if I found a reason, I probably wouldn't be doing it."
The left-hander lasted just 3 2-3 innings and was charged with six runs and 10 hits. He struck out one and walked two as Kansas City lost for the sixth time in seven games.
"Starts where he's gotten through the first inning clean, he generally has real good starts," Royals manager Ned Yost said. "He just struggled to make adjustments today because his stuff wasn't quite as good."
John Danks pitched six strong innings and Paul Konerko added three hits for the White Sox (60-60), who have won eight of 10 to reach .500. They haven't had a winning record since they were 7-6 on April 15.
"Danks was incredibly sharp," Yost said. "His stuff was electric right from the get-go."
Danks was working on a no-hitter before Billy Butler reached on a bad-hop single with two out in the sixth. Butler's roller down the first-base line hopped up and past Lillibridge. Official scorer Bob Rosenberg originally ruled the play an error, but the call was quickly changed to a hit.
"You can't really give him an error on that. Nothing he can do on it," Butler said. "That's a terrible hop. There's bad hops and there's terrible hops, that's a terrible hop."
"When you've got that kind of stuff and it's working, you're going to have a tough day," Butler said. "We battled and got two off him, but it felt like a lot when we got two off a guy like that."
Danks (5-9) allowed four hits, struck out six and walked four, improving to 5-1 with a 2.29 ERA in his last nine starts. The left-hander lost his first eight decisions of the season.
"I know I had (a no-hitter) going from the third inning on," Danks said. "It's one of those things, you have a long way to go. Fortunately, we had enough runs on the board where it wasn't blocking trying to win a ballgame. I was trying to get early contact just like I always do."
Alexei Ramirez had two hits and scored twice as Chicago improved its abysmal home record to 26-33 and moved within four games of AL Central-leading Detroit, which lost 8-5 at Baltimore. The White Sox also won consecutive games in a single series at U.S. Cellular Field for the first time since June 21 and 22 against the Cubs.
The White Sox got off to a fast start against Francis. Ramirez was hit by a pitch with one out and moved to third on Konerko's single. Carlos Quentin then hit an RBI single and Lillibridge drove a 1-2 fastball over the wall in left for his 10th homer.
"Kansas City always gives us a hard time," Lillibridge said. "They play hard and they're a good team. They're always going to grind, they have a lot of talent out there and their pitching's getting a lot better."
The Royals, who own the second-worst record in the American League, now head home for seven games against New York and Boston, the top two teams in the AL.
Top prospect Mike Moustakas went 0 for 3 and is 0 for 34 this season against Chicago.
NOTES: Royals RHP Felipe Paulino is Monday's scheduled starter against the New York Yankees. In his last start, Paulino pitched five innings before leaving with a 3-1 lead after 84 pitches due to back stiffness. ... White Sox C A.J. Pierzynski missed his third straight game with a bruised left wrist. ... White Sox DH Adam Dunn was not with the team for the third straight day to tend to family matters. ... Konerko reached base 10 times on 14 plate appearances during the three-game series.