Smith, 32, is 1-4 with a 3.82 ERA and six saves in 38 relief appearances. In 70 appearances last season, he posted a 3.58 ERA. He had 15 saves in 2014.
In return, Chicago will send Class A right-hander Jesus Castillo to Los Angeles.
The Cubs made one of the biggest splashes of the trade deadline, acquiring closer Aroldis Chapman from the New York Yankees. They also traded for Seattle Mariners left-hander Mike Montgomery and activated longtime closer Joe Nathan.
In other organizational moves, the team recalled righty Spencer Patton on Monday after releasing lefty Brian Matusz, who gave up three two-run home runs to the Mariners on Sunday night. Smith is likely to take Patton's spot when he arrives.
Information from ESPN's Jesse Rogers was used in this report.