Adrian Beltre hit his 32nd home run for Texas.
The Rangers, who clinched the AL West on Friday, fell into a tie with the Boston Red Sox for home-field advantage in the playoffs. Texas (92-64) holds a slight tiebreaking advantage based on division record.
Oakland will be sitting out the postseason and had lost the first five games on this homestand but ended it on an encouraging note behind Cotton (2-0). The rookie right-hander allowed one run, struck out five and faced two over the minimum while quieting the Rangers' potent offense.
Cotton became the first A's pitcher since at least 1913 to start his career with four consecutive starts of five innings or longer while allowing one earned run or less.
Beltre homered off Cotton in the seventh, the 445th of his career. The 37-year-old also singled to move past home run king Barry Bonds on the career hits list with 2,937.
Coming off consecutive shutout losses in the first two games of the series, the A's did all of their scoring in the second to chase starter Colby Lewis (6-4).
Bruce Maxwell blooped an RBI single just past the outstretched glove of shortstop Hanser Alberto. Marcus Semien scored on a groundout and Vogt lined a three-run double to right before Healy's 12th home run, a towering shot that put Oakland up 7-0.
That ended a streak of 23 consecutive scoreless innings by Texas pitchers.
Lewis, who had won four consecutive games at the Coliseum, lasted just 1 1/3 innings and allowed a season-high seven runs and seven hits. The right-hander has lost four consecutive decisions, three since a two-month stint on the disabled list.
Athletics: Sonny Gray will come off the disabled list to make his first start in nearly two months against the Angels on Wednesday. Gray has not pitched in a game since Aug. 6 because of a forearm strain.
Rangers: LHP Martin Perez (10-10) faces the Brewers on Monday when Texas begins its final homestand of the regular season. Perez owns a 2.65 ERA at home, tied for the lowest in the AL among qualifying pitchers.
Athletics: Sean Manaea (6-9) starts against the Angels in Anaheim on Monday. The left-handed rookie has not allowed a run in 11 consecutive innings since missing more than two weeks with a sore back.