Chirinos' sacrifice fly to center scored Adrian Beltre, who led off the inning with a double off Chad Qualls (0-2), moved to third on Kyle Blanks' single and slid in as Jake Marisnick's throw to the plate went to the backstop.
Keone Kela (2-1) pitched the eighth for the win, and Neftali Feliz pitched the ninth for his third save.
Houston scored in the first as Jose Altuve and George Springer drew walks, worked a double steal and Altuve scored on Evan Gattis' sacrifice fly to make it 1-0.
Jake Smolinski tied it in the eighth with a one-out RBI single to center.
Dallas Keuchel, who was named the AL pitcher of the month for April earlier in the day, allowed one run on five hits with a season-high eight strikeouts in eight innings.
Keuchel retired nine straight during one stretch, including striking out the side in the second and third innings, and 14 out of 15 batters between the first and fifth innings.
Other than his struggles in the first inning, Texas starter Ross Detwiler matched Keuchel, allowing one run and four hits with seven strikeouts in seven innings. Detwiler had his second consecutive solid outing after allowing 18 runs over 12 1-3 innings in his first three starts.
Leonys Martin left the game in the fifth with a left wrist sprain and was going for precautionary X-rays. Martin injured his wrist while diving for Altuve's double in the third, but after the trainer and manager Jeff Banister came out to look at him, Martin stayed in. DeShields replaced Martin in center.
Rangers: OF Josh Hamilton (right shoulder) participated in an extended spring training game Monday in Arizona, going 0 for 4. Hamilton was acquired from the Los Angeles Angels on April 27. ... 1B Mitch Moreland hit in the batting cages Monday. Moreland has been on the DL since April 29, retroactive to April 28, for left elbow surgery.
Rangers: LHP Wandy Rodriguez (0-1) will make his third start of the season for Texas in the second game of the three-game series. Rodriguez, who was in the Astros' organization for the 14 years of his career, allowed four runs on five hits with five walks over 4 1-3 innings in his last start Wednesday against Seattle.
Astros: RHP Scott Feldman (2-2) will look to win his third straight decision and continue his string of three consecutive starts Tuesday against his former team. Feldman has allowed seven runs over 19 2-3 innings in his last three starts following a 0-2 start.