Crisp's fifth-inning hit broke a 2-2 deadlock while Brandon Moss and Josh Reddick each added an RBI for the A's, who returned to the playoffs for the first time since 2006.
Jarrod Parker (13-8) hurled six effective innings, allowing three runs on six hits. The rookie right-hander won his fourth straight decision, walking a pair and striking out six.
Sean Doolittle and Ryan Cook each tossed a scoreless relief frame and Grant Balfour nailed down the playoff-clinching victory with his 23rd save.
The win assured Oakland of the second American League wild card berth. The A's still have a chance to grab the AL West crown from the Rangers with two more victories in the season-ending three-game set.
Texas starter Martin Perez (1-4) was touched for four runs on six hits over four frames.
Michael Young and Mark Napoli homered for the Rangers, whose magic number to wrap up the division title remained at one.