Harrison (1-0) scattered four hits over six innings for the Rangers, allowing two walks and striking out three.
Mark Lowe, Robbie Ross and Koji Uehara each contributed an inning of scoreless relief as Texas took the rubber match of a season-opening three-game series.
Gavin Floyd (0-1) absorbed the loss, yielding five runs (four earned) on five hits over 5 2/3 innings.
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