Kurt Suzuki also drove in a run for the Athletics, who have won seven of their last nine games.
David Murphy went 2-for-4 and supplied the Rangers' only run with a solo home run in the fourth inning after Oakland had racked up a 4-0 lead.
Davis drew a bases-loaded walk in the first inning to give the A's a 2-0 advantage, then drove in two runs with a single in the fourth to make it 4-0.
He drove in another run and Suzuki walked with the bases loaded in the sixth to make it 6-1.
Brandon McCarthy (7-3) took the loss after giving up four runs -- three earned -- on four hits in three-plus innings.