The Angels announced the deal Friday.
The A's designated Smith for assignment on Thursday after he struggled in two major league appearances. He gave up four runs while getting just one out Wednesday at Minnesota.
Smith made his major league debut last June for the Detroit Tigers, but was designated for assignment in February.
Smith was assigned to Salt Lake. He was 1-1 with a 2.31 ERA and two saves for Oakland's Triple-A affiliate in Nashville this season.
Smith is a Southern California native who pitched at USC, where he was a teammate of Angels infielder Grant Green.