The Angels have dropped four in a row.
Barton doubled home a run in Oakland's three-run first and the Athletics went on to build a 10-2 lead through seven innings.
He added a run-scoring single in the second, Davis drove in two runs with a single in the sixth, and Cliff Pennington produced a three-run homer in the seventh.
Gio Gonzalez (1-0), Oakland's fifth starter, went six innings to get the victory. He gave up two runs on six hits and struck out six.
Five of the Oakland runs were charged to losing pitcher Matt Palmer (0-1), who was roughed up for eight hits in five innings.
Hideki Matsui and Juan Rivera hit solo homers for the Angels.