Milone (6-3) tamed the Angels on five hits, issuing only one walk while striking out three for the A's, who improved to 22-21 with the victory.
Ryan Cook tossed scoreless eighth and Brian Fuentes turned in a 1-2-3 ninth as Oakland won for only the third time in seven games and claimed the opener of a six-game homestand.
Kila Ka'aihue got the only RBI for Oakland on a third-inning single.
Jerome Williams (4-2) was tagged with the loss, surrendering a pair of runs on five hits in a 6 1/3-inning stint.
Mike Trout drove in the Angels' only run as Los Angeles dropped its third straight.
Celebrity Families of 2014 [PHOTOS]