Ross (2-3) tamed the Angels on five hits over six full innings, issuing one walk while fanning two batters in breaking a personal three-game losing streak.
Grant Balfour, Jordan Norberto and Ryan Cook each tossed a scoreless relief frame to complete the shutout, giving the A's a victory in the first contest of a two-game series at Angels Stadium.
Seth Smith provided a two-run double and Josh Reddick smacked a solo homer in Oakland's triumph.
Dan Haren (1-4) absorbed his third straight loss for Los Angeles, surrendering four runs on six hits in a six-inning effort.
The Angels played without star outfielder Torii Hunter, who was placed on the restricted list for unspecified personal reasons.
The move came after reports surfaced that Hunter's 17-year-old son had been arrested on sexual assault charges in Texas.
Celebrity Breakups and divorces of 2014 [PHOTOS]