Andy Ibanez, the Rangers' No. 12 prospect according to MLBPipeline.com, also hit a home run after being brought over from the Minor League side.
Rangers right-hander Yu Darvish, almost two years removed from Tommy John surgery, looked sharp in his first spring outing, throwing 1 2/3 shutout innings.
Darvish threw 37 pitches and touched 96 mph a few times while sitting at 93-94. He gave up one hit, walked two and struck out two.
"It was a good first day to get him back out there," Rangers manager Jeff Banister said. "We are comfortable with what we saw. Obviously the two walks, that's not where we would like to be. But he got out there and got his work in."
Zimmer, who is coming off thoracic outlet syndrome surgery last August, wobbled a bit through 1 1/3 innings. He gave up four hits.
Zimmer's velocity touched 95 mph in the first inning, but it tapered off to 90-91 in the second. He walked one and struck out one.
"I thought he was OK," Royals manager Ned Yost said. "But I don't put a lot of stock into first outings in Spring Training."
Chase Vallot, the Royals' No. 7 prospect, hit a two-run triple as Kansas City rallied for three runs in the ninth.
Up next for Rangers: Listen live for free to an exclusive webcast as Eddie Gamboa pitches against the Brewers in Surprise, while Chi Chi Gonzalez faces the Indians in Goodyear on Monday in split-squad action. Both games start at 2:05 p.m. CT. Gamboa is a knuckleballer, who was acquired from the Rays on Feb. 14. Both are candidates for the fifth spot in the rotation.
Up next for Royals: The Royals will see former teammate Jarrod Dyson in action for the first time since he was traded in the offseason when they face the Mariners on Monday live on Gameday Audio at 2:10 p.m. CT in Peoria. Left-hander Jason Vargas, who could be the Royals' No. 3 or No. 4 starter, gets the ball.