Rios, obtained from the Chicago White Sox in a trade Friday, proved to be the difference as the Rangers came back from a 4-3 deficit in the eighth for a sixth straight win and grabbed first place in the American League West.
A.J. Pierzynski doubled with one out against Lucas Harrell (5-13) and Rios followed by sending a 3-0 pitch into center field that eluded Houston's Brandon Barnes and rolled behind him for a score-tying triple.
Mitch Moreland then hit a grounder to second baseman Jose Altuve and Rios broke for the plate. The throw home was in time but catcher Jason Castro dropped the ball for an error, allowing the decisive run to score.
Elvis Andrus socked a two-run homer and Pierzynski added a solo shot in the victory.
Chris Carter went 2-for-3 with a solo homer while Altuve and Matt Dominguez each drove in a run for the Astros.