Rodriguez, 37, batted .251 with eight homers and 34 RBI in 93 games as Houston's starting catcher this year. He will be the backup catcher to Taylor Teagarden, with regular starter Jarrod Saltalamacchia sidelined with a shoulder injury.
"I'm going to move forward. It's a good trade," Rodriguez said. "I'm looking forward to going there and to do my best to try and help them win."
Rodriguez spent the first 12 seasons of his 19-year career in Texas, where he won the 1999 American League MVP after batting .332 with 35 homers and 113 RBI.
A 13-time Gold Glove winner and 14-time all-star, Rodriguez has compiled a .299 average with 303 home runs and 1,251 RBI throughout his career.
He caught his 2,227th game this season, surpassing Carlton Fisk for most games played by a catcher in MLB history.