Darvish gave up a leadoff single to shortstop Alcides Escobar, then retired the next three hitters, including a strikeout of designated hitter Kendrys Morales. Darvish, who was slated to throw 35 pitches, threw 12 pitches, 10 for strikes.
"Today,when I was throwing in the bullpen, it started to feel like it was tightening up," Darvish said, "so from then on I started to throw without putting to much effort."
The Rangers took the lead 1-0 in the second inning when Jared Hoying beat out a swinging bunt and later scored on a single by Robinson Chirinos.
In the third, the Rangers nicked Royals right-handed prospect Miguel Almonte for two runs, thanks to a fly ball to left by Ryan Rua that Royals' outfielder Reymond Fuentes lost in the sun and missed by 30 feet. It was scored a triple.
Almonte, though, showed off a terrific curve and a fastball that touched 98 mph. He struck out the side in his one inning of work.
The Royals got on the board in the fourth on consecutive singles from Jarrod Dyson, Eric Hosmer and Morales. Jorge Bonifacio later singled in Hosmer, who barely beat the throw to home plate.
Former Royals' outfielder Carlos Peguero ripped a run-scoring double to right for the Rangers in the sixth, making it 4-2.
Left-hander Brian Flynn, fighting for a possible spot in the bullpen for the Royals, threw the first two innings and gave up three hits and a run.
Up next: Nick Tepesch and Anthony Ranaudo will be on the mound for the Rangers on Friday when they face the Giants at 2:05 (CT) p.m. at Surprise Stadium. They are among those who are competing for the fifth spot in the rotation. Ranaudo was acquired from the Red Sox for Robbie Ross in the offseason. The game can be heard in an exclusive webcast on rangers.com.
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @jflanagankc. This story was not subject to the approval of Major League Baseball or its clubs.