Darvish (2-1) was in midseason form for the second straight outing. He gave up five hits, struck out four and walked one.
"Just another great start from him," Chicago manager David Ross said. "I thought he really took off after we took the lead."
With the score 2-1, the Royals' best chance to tie the game came in the sixth when Whit Merrifield was on third with one out. Darvish struck out Ryan O'Hearn and retired Salvador Perez on a grounder, ending the threat.
The Cubs capitalized on one bad stretch for Royals starter Kris Bubic, who set down his first nine batters, including four strikeouts, before Bryant singled in the fourth.
Rizzo followed with a walk, and then a bloop hit by Baez tied it 1-all. Willson Contreras grounded into a double play, but Rizzo scored to give the Cubs a 2-1 lead.
"The guys had great at-bats and baserunning," Ross said. "The attention to detail has been really good."
Kansas City (3-10) started the scoring in the third when a double off the left-field wall by Maikel Franco was followed by Nicky Lopez's single, but a potential big inning was thwarted when Merrifield grounded into a 1-4-3 double play.
The Cubs took advantage of two fielding mistakes in the eighth and ninth, tacking on four runs to make it 6-1.
Bubic (0-2), making his second career start, went six innings and gave up two runs. He struck out six.
"That was exactly what we were looking for," Royals manager Mike Matheny said. "His changeup was really good and breaking ball was good, too."
Royals: Before the game, OF Bubba Starling and RHP Glenn Sparkman were placed on the injured list. Starling was batting .211 with three RBI. Sparkman had a 5.40 ERA and allowed nine hits in four appearances. Kansas City recalled catcher Meibrys Viloria and RHP Scott Blewett.
Royals RHP Brad Keller makes his season debut Thursday night after returning from the IL due to COVID-19. RHP Tyler Chatwood (2-0, 0.74 ERA) will start for the Cubs. "I feel pretty good," Keller said Saturday. "Whenever they tell me, I'll be ready."
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