Neal Cotts (8-3) recorded the decision in the must-win final regular-season game of the year for the Rangers (91-71), who will host Tampa Bay Monday in the play-in contest.
The winner of that tilt will then go on to face Cleveland in Wednesday's American League Wild Card Game.
Rangers Yu Darvish allowed two runs on four hits over 5 2/3 innings as surging Texas reeled off a seventh straight victory.
Jason Vargas (9-8) was hung with the loss, yielding four runs (three earned) on seven hits and two walks over 6 1/3 innings for the Angels (78-84).
Mike Trout homered as Los Angeles finished the season with a fourth straight loss.
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