Oakland scored single runs in the fifth, sixth and seventh innings but the Rangers answered in the bottom of each inning, finally taking the lead on Kinsler's hit.
David Murphy led off the Texas seventh with a double off reliever Ryan Cook but was out on a field's choice, with Robinson Chirinos reaching. Sean Doolittle replaced Cook but allowed a single to A.J. Pierzynski. After another out, Leonys Martin singled, loading the bases.
Kinsler then lined a 3-2 pitch into center, allowing Chirinos and Pierzynski to score and saddling Doolittle (3-2) with the loss.
Joe Nathan, who got his 22nd save, struck out two but also allowed two hits. Josh Donaldson tried to score from first on the second one but was thrown over in a strong relay from center fielder Craig Gentry to shortstop Elvis to Pierzynski at catcher to end the game.
That allowed Tanner Scheppers to improve to 5-0 with 1 2/3 innings of one-hit relief.
Oakland scored the first run of the game on a Jed Lowrie fifth-inning single but Geovany Soto tied the score for Texas with a home run in the bottom of the inning. Chris Young walked with the bases loaded in the sixth, putting the As' ahead 2-1, a lead that was erased when Kinsler scored on a wild pitch.
Oakland went up 3-2 in the seventh on Josh Donaldson's bases-loaded infield hit.