Texas is 1-4 on its homestand and 5 1/2 games behind AL wild-card leader Boston. The Rangers have their largest deficit for a playoff spot all season with only 18 games left.
David Murphy's leadoff homer in the fourth snapped the Rangers' 22-inning scoreless streak, their longest in five years, but they were down 4-1 and didn't score again.
Craig Breslow (7-7) worked two scoreless innings in relief for the last-place Athletics, who have won seven of nine.
Boston won 4-1 on Tuesday night in the opener of a three-game series against the AL West-leading Los Angeles Angels, who have a six-game division lead over Texas. After playing at Fenway Park, the Angels will be in Texas this weekend.
Michael Young returned to the Texas lineup as the designated hitter after missing two weeks because of a strained left hamstring, but the All-Star third baseman left after he flied out his only at-bat in the first inning.
The Rangers said there was no physical problem but that Young, who reported that he felt OK and just wanted another day or two before playing full-time again.
Davis, who finished with three hits, put the A's ahead to stay when he reached on a fielder's choice in the first, stole his 37th base and went to third on catcher Ivan Rodriguez's throwing error before scoring on Ryan Sweeney's groundout.
Davis drew a bases-loaded walk in the second off Brandon McCarthy (7-3), had a two-run single in the fourth and another run-scoring single in the sixth. Davis was caught stealing after a leadoff single in the eighth.
McCarthy, who had won his two previous starts after nearly three months on the disabled list (stress fracture in right shoulder blade), gave up four runs while pitching into the fourth.
Oakland starter Edgar Gonzalez was pulled after allowing the Nos. 8 and 9 hitters to reach base starting the fifth.
Rangers hard-throwing rookie Neftali Feliz struck out the last three batters with the bases loaded in the sixth, after giving up two runs on a leadoff triple, two walks (one with the bases loaded), a hit batsman and Davis' single. Feliz had allowed only two runs over 23 2/3 innings his first 13 major league appearances.
Young got hurt running out a grounder Sept. 1, and had repeatedly said since that he was targeting this weekend's series against the Angels for his return. But he tested himself by running the bases before the game and was back in the lineup several days earlier than expected.
Notes
Elvis Andrus went 0-for-2 with two walks, ending his Rangers' rookie record 16-game hitting streak. … Texas went 25 consecutive innings without a run in September 2004, when they were last in playoff contention. The team record of 28 was set in 1972. … Boston's victory officially eliminated Oakland from postseason contention. It came in the 144th game, one later than last season, which was the earliest elimination for the A's since 1997. … The Rangers had three errors.