The win was only the fourth for the Astros in their last 20 games.
Justin Maxwell produced a two-run home run for Houston in the first inning, but the Astros faced a 3-2 deficit heading into the ninth.
Tyler Greene and J.D. Martinez both delivered one-run singles off Aroldis Chapman (5-5) before Dominguez hit an 0-1 pitch over the wall in left. Dominguez has appeared in 75 career games scattered over two seasons.
Hector Ambriz (1-0) recorded the final two outs in the eighth inning and was credited with his first major league victory. His only other two decisions had been losses in 2010 while he was with Cleveland.
Wilton Lopez had the benefit of a double play in the bottom of the ninth while collecting his third save.