Garza (4-5) allowed five hits and a walk while striking out five, pinpointing 70-of-97 pitches for strikes.
Joe Nathan closed out the win with his 40th save as the Rangers finished the night a half-game behind Cleveland for the second and final American League wild card spot.
Royals starter Jeremy Guthrie (14-12) was reached for three runs on seven hits over six innings.
Eric Hosmer's solo homer was the only offense for Kansas City.
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