The Rangers find themselves just one game behind co-leaders Tampa Bay and Cleveland in the chase for the American League's two wild card berths.
Unless the Rays or the Indians lose both their remaining regular-season games, the best Texas can hope for is to earn a spot in a playoff game for one of the wild card berths.
The Rangers won their fifth in a row, coming through with a key seventh inning against J.C. Gutierrez (1-5).
Rios then singled to right to bring in Kinsler with the run that snapped a 3-3 tie. Rios stole second and eventually scored an insurance run on a single by A.J. Pierzynski.
Neal Cotts (7-3), the third of five Texas pitchers, earned the victory by striking out two in a perfect seventh inning. Joe Nathan retired the side in order in the ninth on just seven pitches for his 42nd save.