Michael Young walked and Hank Blalock singled with two outs before Rodriguez doubled to left field off Jose Santiago (0-1), putting the Rangers ahead to stay and sending the Indians to their ninth defeat in 10 games.
Rodriguez had two hits and is hitting .555 in his last six games. But the shortstop made his first error of the season on a ball hit by Tim Laker in the second inning.
Todd Van Poppel of Texas made his first start since the 2000 season and allowed five runs -- three earned -- and five hits in four innings.
Five relievers followed for Texas, with Jay Powell (1-0) following Van Poppel and pitching two scoreless innings for his first win since July 21.
Notable deaths of 2014 [PHOTOS]