Guthrie (1-0) scattered five hits and walked one for Kansas City, which took its first lead of the season with three runs in the fifth inning and made it stick.
“Guthrie was phenomenal today,” Royals manager Ned Yost said. “He really executed his pitches, changing speeds, moving the ball in and out. Really did a nice job keeping them off balance.”
The Royals scored just three runs in the first two games of the season and lost both, including 1-0 in the season opener.
Greg Holland worked the ninth for his first save.
There was also a lot to like about Guthrie's counterpart — Chicago right-hander Gavin Floyd.
Floyd (0-1) gave up four hits and two runs in six innings. He walked one and struck out five.
“I thought he pitched great,” said White Sox second baseman Gordon Beckham, who went 4 for 4. “The one inning that got away from him a little bit I thought he still made good pitches and just hung a couple.”
Guthrie, meanwhile, allowed a run in the bottom of the fifth against an aggressive-swinging White Sox team that won the series' first two games.
“Any time a team's more aggressive you've got to make sure you make better quality pitches knowing they're going to try to get that ball in play and drive it,” Guthrie said. “So that's kind of the case it was this whole series, they were aggressive and today was no different.”
The Royals scored three runs after the first out in the fifth. First baseman Eric Hosmer walked, advanced to third on Francoeur's single to right on a hit-and-run and scored when center fielder Jarrod Dyson grounded out.
Francoeur advanced to second on the play and came home on Chris Getz's single to center field and an error by Alejandro De Aza. Getz, who advanced to second base after the error, scored on Alex Gordon's RBI single to right.
“I thought that (hit-and-run) was huge and then Getzy to follow through and drive me in it gave us a little bit of breathing room,” Francoeur said.
Yost said the outburst was encouraging.
“You know, it was good today,” he said. “We got the hit and run, Dyson got the first run in, everybody relaxed, and boom, boom, we've got a three-run lead. It's coming.”
Chicago got a run back in the bottom half. Tyler Flowers was hit by a pitch from Guthrie with two outs, advanced to third on Beckham's second hit of the game and scored on De Aza's single to right-center before Jeff Keppinger struck out to close the inning.
Beckham singled to right in the seventh off reliever Aaron Crow to put runners at first and second with one out. But Dyson made a running grab of De Aza's long fly and the Royals got out of the inning when Escobar grounded to second.
Beckham got his fourth hit in the ninth, a two-out single that advanced pinch-runner Conor Gillaspie to second. But De Aza grounded to first to end the game.
Chicago had runners on first and third with just one out in the third after Beckham lined a single to right to advance Flowers. But De Aza, the leadoff batter, grounded into an inning-ending double play.
Kansas City catcher Salvador Perez lined a one-out double to left off Floyd in the second inning but failed to advance.
Guthrie was 5-3 in for the Royals last year following his July acquisition from Colorado. Kansas City was 10-4 in his 14 starts.
“You've got to tip your hat to Guthrie for what he did,” White Sox manager Robin Ventura said. “He didn't give us too many chances. When we had a chance there at first and third and he got the ground ball.
Except for the struggles in the fifth, Floyd was solid. He threw 94 pitches — 66 for strikes. He was 12-10 last year, his first winning season since 2008 but fifth straight with double-digit wins.
Four Kansas City pitchers combined to strike out 11. Dayan Viciedo struck out four times and Adam Dunn struck out three.
NOTES: Flowers is only the second White Sox catcher to homer in his first two games. Sherm Lollar had a pair in 1958. ... The White Sox continue a six-game home stand at 7:10 p.m. CDT Friday with the first in a three-game series with Seattle. Chicago sends left-hander Jose Quintana against right-hander Blake Beavan. ... The Royals open a three-game set in Philadelphia at 4:05 p.m. Friday as right-hander Wade Davis pitched against Phillies' right-hander Kyle Kendrick. It's also the first starts for each. ... Kansas City's Francoeur is looking forward to the unusual early season series with a National League team. “I think it's intriguing,” he said. “I remember when I looked at our schedule thinking `We're going to Philly the second series of the year?' but I think it's going to be pretty cool and it will be a good experience.” The Royals lost two of three in their only previous visit to Philadelphia in 2004. ... Chicago's first interleague series starts next Wednesday with a three-game set in Washington.