Straily (2-2) allowed only two hits and didn't issue a walk over seven scoreless frames, striking out five. The right-hander broke a personal four-start winless streak with the decision.
Sean Doolittle blanked Texas in the eighth and Grant Balfour tossed a 1-2-3 ninth for his ninth save as Oakland clinched its three-game series with a fifth straight win.
Yoenis Cespedes supplied the game's only run with a third-inning solo homer.
Texas starter Yu Darvish (7-2) absorbed the hard-luck loss, yielding just one run on five hits over six full frames.