Rasmussen, 23, is a native of Arcadia and played collegiately at UCLA after starring at Pasadena Poly High School. In 2007, the Dodgers selected him in the 27th round of the amateur draft, but he decided to attend UCLA instead.
In three minor league seasons, Rasmussen is 20-21 with a 3.88 ERA in 60 games (53 starts), including 4-4 with a 4.80 ERA in 11 games (10 starts) at Double-A Corpus Christi in 2012.
Ely, 26, was acquired in a December 2009 trade with the Chicago White Sox and went 4-13 with a 5.70 ERA in 25 games (19 starts) over three seasons in Los Angeles.
The Dodgers now have 39 players on their 40-man roster.