Griffin, who spent a month on the disabled list this year, is 6-0 in his 11 starts. After throwing 107 pitches while giving up six hits, striking out six and walking no one, he was not given a chance for what would have been his first complete game and shutout in the majors.
Sean Doolittle relieved Griffin in the ninth and gave up a homer to Albert Pujols for the Angels' lone run. Ryan Cook came on to record the final out and earn his 14th save.
The win allowed Oakland to keep pressure on division-leading Texas, which also was victorious Wednesday and remained three games in front of the Athletics in the American League West. Beginning with its game on Aug. 16, Oakland has compiled a 21-5 record.
Oakland scored a run on a first-inning fielding error by shortstop Erick Aybar and Yoenis Cespedes hit a solo homer in the sixth.
The other two A's runs came in the eighth, one on a double by Derek Norris and another on a single by Coco Crisp.
Ervin Santana (8-12) gave up only one earned run in six innings, but took the loss.