Rodriguez, a 27-year-old, four-year veteran, has a 10-15 career record with a 5.18 ERA for Cleveland and Texas. He was 2-3 with a 5.53 ERA in 12 games in 2005.
Padilla was originally traded to Texas for a player to be named earlier this off-season.
Padilla, 28, was 49-45 overall in Philadelphia and represented the final piece of the four-for-one trade that sent Curt Schilling to Arizona in July 2000. Philadelphia also received Travis Lee, Omar Daal and Nelson Figueroa in the trade.
Padilla joins a Rangers rotation that lost Kenny Rogers, who signed a two-year contract with the Detroit Tigers last Monday. In 2005, Texas' pitching staff finished 26th in Major League Baseball with a 4.96 earned run average.