The Royals' record is 22-6-1; the Dodgers are 11-17-3.
Greinke said his elbow felt fine after pitching in his first Major League exhibition game since March 1 and he should be ready to start the season on time.
"I should be. I think so," he said. "I'll definitely be there, if I can build up endurance. I need some improvement."
Greinke's return to the Dodgers' working ranks lasted fewer than four innings. His departure came after he began the fourth with three straight walks.
At the time, the Royals held a 2-1 lead, but the inning got even messier under the stewardship of reliever Matt Guerrier. Chris Getz started Guerrier's night with an RBI single, and Alex Gordon's swinging bunt to third base led to a Juan Uribe throwing error and two more runs. Mike Moustakas finished the four-run rally with another RBI single.
Greinke was charged with five runs on six hits and the three walks. Back-to-back singles by Gordon and Alcides Escobar, plus Salvador Perez's double cost Greinke two runs in the first inning.
Royals starter Wade Davis went five innings and gave up two runs (one earned) on four hits and three walks. He is 4-0 in four starts this spring.
Dodgers manager Don Mattingly was ejected after the top of the seventh inning for arguing with plate umpire Anthony Johnson after Matt Kemp's check-swing third strike. Kemp went 0-for-3, lowering his spring average to .171, but he did pick up his sixth RBI with a sac fly off Davis in the fifth, and reached on a run-scoring error in the third.
Dodgers reliever J.P. Howell struck out four Royals in a six-batter span in the fifth and sixth innings, and closer Brandon League worked a scoreless eighth inning.
Up next: One day after scouts watched Chris Capuano in a Minor League game against the Indians, they likely will return to watch Aaron Harang start for the Dodgers on Tuesday night against Colorado. The Dodgers are willing to deal at least one of their three extra starters, Ted Lilly being the third, as it would help them keep non-roster reliever Kevin Gregg.
The 7:05 PT nightcap is the Dodgers' last home game of the Cactus League schedule and will be available free on MLB.TV.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.