Floyd also struck out five as he tossed his fourth straight quality start.
"He was spotting his fastball to both sides of the plate, keeping it down, good slider -- really good slider," Royals manager Ned Yost said of Floyd. "That's what kept our right-handed hitters at bay. He was just executing quality pitches all night long."
Royals left fielder Alex Gordon, who went 1-for-4, said Floyd's slider set him apart on Friday.
"We knew he had a good slider. I don't know if it's his go-to pitch, but it seemed like it was tonight," Gordon said. "It seemed to be working for him and he was throwing it when we were looking fastball. He kept us off-balance."
The Royals had an early scoring opportunity in the second, when Mike Moustakas singled and Floyd issued his only two walks of the night to load the bases with one out.
Shortstop Alcides Escobar fouled off Floyd's first pitch before hitting a soft grounder back at Floyd, who threw home for the force out. Humberto Quintero also proved a bit too eager, as the Royals catcher swung at the first pitch and hit another soft grounder back to the mound for the final out.
"They're a tough lineup and I think that was what was so impressive about Gavin tonight," White Sox manager Robin Ventura said. "That's a young team, but they have a lot of hitters on that young team. That's a tough lineup to go through."
Escobar, Dyson and Gordon all singled in the eighth, loading the bases with one out for Billy Butler. Floyd struck out Butler in his final act of the night, as White Sox reliever Matt Thornton came in and got Eric Hosmer swinging to end the threat.
"Give him credit. He pitched a great game and we didn't do a very good job of hitting," Gordon said. "That's just a bad combination."
Royals right-hander Felipe Paulino took the loss in his second start of the season, keeping it close through five innings until White Sox right fielder Alex Rios' two-run triple in the sixth cemented the game.
White Sox second baseman Gordon Beckham led off the sixth with a double off the wall and, with Adam Dunn on first, a fielder's choice put runners on first and third with two outs. Rios -- who had struck out twice previously -- laced a 3-1 fastball into center field to give the hosts a 4-0 edge.
"That was the killer,' Yost said. "You get [behind] 3-2, [Paulino had] struck him out twice before, still throwing the ball good, just got to 3-2, and he still needed to make a pitch."
The result wasn't what the Royals were hoping for, but Paulino said he located the pitch perfectly.
Paulino (1-1), who tossed six scoreless innings in his season debut against the Yankees, was solid until the sixth on Friday. The righty made one mistake -- a solo home run by Dunn in the first -- and got out of a third-inning jam by allowing only one run thanks to a run-saving, diving catch by Gordon in left field.
Beckham singled home Alejandro De Aza, who led off with a single before stealing second, and Dunn followed with another rocket, this time into the right-field corner. Beckham would have scored, but the ball bounced over the wall for a ground-rule double, keeping Beckham at third.
Paulino fell behind White Sox first baseman Paul Konerko, 3-1, and came inside with a two-seam fastball that broke Konerko's bat as he hit a soft liner into left field that Gordon dove for and caught.
Gordon's catch ended an inning that began with an impressive play deep in the hole by Escobar, who back-handed Dayan Viciedo's grounder and jump-threw to Hosmer, who scooped a two-hopper for the inning's first out.
"I was playing him deep, with his power, and I saw the broken bat and came in right away. I was just trying to help out any way I could," Gordon said.