Castro's solo blast in the third inning gave the Astros a 3-1 lead, but the Royals responded with a three-run clout from Miguel Tejada in the next half-inning.
In the fourth, the Astros put Carlos Pena and Jimmy Paredes aboard with one out and Dominguez followed by bashing a pitch from Jeremy Guthrie into the center-field bleachers, putting Houston ahead to stay.
Dallas Keuchel (1-1) earned the victory despite yielding four runs on nine hits over 6 2/3 innings.
Guthrie (5-2) was tagged for six runs on eight hits over five full frames for Kansas City, which fell in the opener of a three game set.
Notable deaths of 2014 [PHOTOS]