The Royals loaded the bases with none out on three walks, but rookie Wil Myers rapped into a third-to-home double play that left runners at first and second with two out. Kouzmanoff batted for Jarrod Dyson and belted Kelvin De La Cruz's 2-0 pitch into the left-center field gap, scoring Mike Moustakas and Myers to end the game.
The comeback came after the Rangers' Elvis Andrus was 3-for-3, including a run-scoring triple, and made a diving stop at shortstop, as the Rangers took an early 6-1 lead.
Adrian Beltre also was 3-for-3 with a double and three RBIs and Ian Kinsler contributed two singles. The top four hitters in the Rangers' lineup -- Kinsler, Andrus, Josh Hamilton and Beltre -- were a cumulative 9-for-12 with four RBIs in the five innings they played.
"That's too bad it doesn't count," Andrus said. "But I feel good at the plate and it's good to get my fielding going. It was a good day today."
Alex Gordon belted a two-run homer to right field for the Royals in the fifth inning against left-hander Mitch Stetter. The drive followed Alcides Escobar's triple off the center-field wall.
"With Esky on third, I was just trying to get him in any way I could, and that's usually when something like that happens," Gordon said. "I was just trying to stay simple with a tough lefty like him and he made a little mistake, and I took advantage of it."
Left-hander Bruce Chen, the Royals' top winner in each of the last two seasons, gave up five runs on seven hits. He was pulled after 1 2/3 innings and ran up 45 pitches in his first start.
Kinsler, Andrus and Beltre each had two hits against Chen. Andrus ripped an RBI triple and Beltre had two run-scoring singles. A first-inning run scored on a throwing error by second baseman Johnny Giavotella.
Andrus' diving stab of Billy Butler's hot shot started a double play that snuffed out a Royals threat in the third inning. Andrus added his third hit, a single, in the fourth against left-hander Mike Montgomery and scored on Beltre's third hit, a double off the top of the right-field wall.
After the Royals' starters left the game, Derrick Robinson smacked a single with the bases loaded for two runs in the sixth against lefty Neal Cotts.
Rangers up next: Mike Napoli will be back behind the plate for the Rangers when they play the Arizona Diamondbacks at 2:05 p.m. CT on Tuesday at Salt River Fields. Napoli started the Cactus League opener on Sunday and had no problems with his high left ankle sprain. He will be catching Rangers starter Matt Harrison.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.