The win trimmed the Rangers lead over second-place Oakland in the AL West to four games.
Kottaras led off the 10th by pouncing on a 3-2 fastball from Mark Lowe (0-2) and sending it into the right-field stands.
Evan Scribner (1-0) pitched the ninth and earned his first Major League win. Grant Balfour picked up his 21st save. The A's staff held the Rangers scoreless after the first inning.
The A's staff recorded seven strikeouts in the game, but the Rangers combined for 15 strikeouts.
Celebrity Families of 2014 [PHOTOS]