The game was tied 2-2 with none out in the fifth when Michael Choice reached first on a throwing error by Rangers third baseman Adrian Beltre.
Crisp followed by smacking a pitch to left field that snuck into the corner for a two-run shot.
After Rangers Manager Ron Washington protested the ball was foul, umpires reviewed the call and videos showed it just glanced off the foul pole.
Yoenis Cespedes also homered for the surging A's, who racked up their seventh win in eight games and forged a tie with Texas atop the American League West in the first of a key three-game series.
Dan Straily (8-7) allowed two runs on three hits over five innings to claim the win.
The A's bullpen followed with four shutout innings, capped by Grant Balfour's 36th save.
Derek Holland (9-7) absorbed the loss, yielding four runs (three earned) on five hits over 4 2/3 innings.
David Murphy smacked a two-run homer for Texas.