Chicago right-hander Gavin Floyd (0-1) gave up four hits and two runs in six innings. He walked one and struck out five.
Royals catcher Chris Getz singled home Jeff Francoeur in the fifth.
Gordon Beckham was 4 for 4 for Chicago, matching a career high set on May 12, 2012, at Minnesota.
Greg Holland worked the ninth for his first save.
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 and scored when center fielder Jarrod Dyson grounded out.
Francoeur advanced to second on the play and came home with 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.
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 advance 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.
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.