Los Angeles had won Saturday's game, 3-2, thanks to a Maicer Izturis RBI single in the ninth. But on Sunday, it was Texas' turn to pull out the win with late-inning heroics.
Angels reliever Rich Thompson (1-2) began the eighth by walking Adrian Beltre and David Murphy, and Mike Napoli singled to load the bases. Hisanori Takahashi came in to replace the ineffective Thompson and faced Davis, who singled to right to plate Beltre.
Despite still having the bases loaded and nobody out, the Rangers didn't score more runs. Takahashi used a force out at the plate, a pop-up to third, and a fly ball to left to keep it a 5-4 game.
But the Angels couldn't plate the tying run against Neftali Feliz in the ninth. Howie Kendrick, who walked with one away, was caught stealing as Alberto Callaspo struck out.
Feliz earned his eighth save for Texas, while Beltre slugged a three-run homer and Ian Kinsler had four hits.
Erick Aybar extended his career-high hitting streak to 13 games by going 3- for-4. He homered, doubled and drove in three for the Angels, who have dropped their last two series.
They lost Sunday's contest after not taking advantage of early scoring opportunities against Rangers starter C.J. Wilson, who began the game by walking Peter Bourjos and allowing a double to Aybar.
But Los Angeles got only one run out of the situation, when Torii Hunter grounded out to short.
They Angels put multiple runners on in the second and third innings, but came up empty both times.
Meanwhile, Texas had wasted a bases-loaded chance in the first inning before cashing in during the third. Elvis Andrus and Kinsler hit back-to-back one out singles to set up the opportunity before Michael Young flied out.
Beltre then turned on Ervin Santana's 2-1 inside fastball, slugging it over the left field wall for a 3-1 lead.
Davis added his solo shot in the fourth, when he strong-armed a low fastball over the left-center field fence -- the opposite way for the first baseman.
Those were the only runs Santana allowed in a seven-inning start, during which he gave up eight hits, walked one and struck out three.
Wilson lasted just five-plus innings after the laborious start. He gave up a second run in the fifth, when Aybar homered to left-center to make it a 4-2 game, and was pulled after Callaspo and Mark Trumbo began the sixth with singles.
Beltre made matters worse for new pitcher Brett Tomko by failing to get an out on Jeff Mathis' sacrifice bunt. Tomko retired the next two batters but couldn't escape the jam, as Aybar lined a two-run single to right field to make it a 4-4 game.
Tomko pitched a scoreless seventh before getting into some trouble in the top of the eighth, when Alexi Amarista tripled with one out. But Amarista was caught in a rundown after Bourjos hit a ball to Beltre at third, and Mark Lowe (1-0) got the final out of the inning.
The Angels were 1-for-11 with runners in scoring position and stranded 11...The Rangers batted 4-for-13 with RISP and left nine on base...Wilson allowed six hits, walked four and struck out two in his start.
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