The 26-year-old left-hander is 27-32 with three saves and a 3.73 ERA in 92 games over six major league seasons with Oakland and Colorado.
As a member of the Rockies last season, Anderson was limited to eight starts because of a broken left index finger and lower back surgery. He went 1-3 with a 2.91 ERA.
Anderson was originally selected by Arizona in the second round of the 2006 first-year player draft.
To make room on the 40-man roster, the Dodgers designated infielder Erisbel Arruebarrena for assignment Wednesday.