The Angels announced the deal Monday for Diaz, who was designated for assignment by Chicago last week.
Los Angeles also designated left-hander Jason Gurka for assignment.
Diaz had a 4.98 ERA in 36 appearances with the Cubs over the past two seasons. The Dominican prospect showed talent and a 98 mph fastball but gave up eight homers this season.
Angels general manager Billy Eppler has demonstrated a knack for discovering capable pitchers on other teams' scrap heaps. He added relievers Blake Parker, Yusmeiro Petit, David Hernandez and Bud Norris and starter Parker Bridwell at minimal expense in the past year alone.