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Alejandro De Aza and Gordon Beckham also went deep for AL Central-leading Chicago, which maintained a three-game advantage over Detroit and improved to 6-10 against the Royals. Chicago had lost seven of eight against Kansas City.
Floyd (10-10) gave up two runs in the first inning and nothing after that. He allowed five hits, walked one and struck out three, throwing 52 strikes in 78 pitches.
The 6-foot-6 right-hander was making his second start since coming off the disabled list after being sidelined with an elbow flexor strain. He also missed time earlier in the year with right elbow tendinitis.
"I felt good out there," he said. "I just wanted to make as few pitches out there and let the defense work. It's great especially after sitting for 15 days and more than that before. It's nice to get back out there and just be part of the team and contribute."
Floyd reached double-figures in victories for the fifth consecutive season as the White Sox (81-66) moved 15 games above .500.
"We're no way content," Floyd said. "We're content in the place that we are, but there's still a lot of baseball left. It's fun to be in the position we're at. We've just got to keep pushing."
De Aza drove Luke Hochevar's second pitch over the wall in right for his eighth homer of the season. De Aza had been slumping, hitless in his previous nine at-bat and 2-for-21 in his previous five games.
First-year manager Robin Ventura did not play the outfielder in the past two games, but thought the timing was right to get him back in the lineup.
"I know what I'm doing," Ventura joked. "He looked out of synch in Minnesota."
Hochevar (8-14), who is 1-5 in his past nine starts, allowed only one single the next four innings before Beckham led off the sixth with his 16th home run. Beckham is hitting .333 with four home runs and 14 RBI in his past 17 games.
"He's in a better position to hit," Ventura said. "He's more confident. He's getting his hands back and getting through it. He's hit some homers, but for me he's keeping the ball on a line a little bit more than he has all year long."
The White Sox upped their home run total to 198, which ranks second in the majors. They have 13 home runs in their past 14 road games.
"I'm not going to be critical how we get them," Ventura said. "This is a big ballpark. You've got to find a way to get some runs other than just sitting back and waiting for a home run. I'm glad we hit them, but this isn't a ballpark you can sit back and do that."
Rios' 24th home run came with one out in the seventh and matched his career high. Rios also played a key role in Chicago's 5-4 victory against Detroit on Monday when he broke up a potential double play with a hard slide that led to an errant throw that allowed the tying and go-ahead runs to score.
Hochevar allowed seven hits in seven innings. He struck out four and walked none.
Kansas City got off to a fast start when Billy Butler hit a two-run single in the first inning to give him a career-high 97 RBI.
The Royals did not have a baserunner after Jeff Francoeur's fourth-inning double.
"He [Floyd] settled in really nice," Royals manager Ned Yost said. "They retired our last 16 batters. We had trouble centering anything up against them."
- Royals RHP Luis Mendoza has strep throat and will miss his scheduled Wednesday start against the White Sox. LHP Bruce Chen will be moved up a day, while RHP Jeremy Guthrie will start Thursday.
- The White Sox announced LHP Francisco Liriano will start the Thursday series finale against the Royals.
- Royals LF Alex Gordon threw out A.J. Pierzynski trying to stretch a single into a double in the fifth inning. Kansas City leads the majors with 44 outfield assists.