Harrison (9-3) allowed an RBI double by Chase Headley in the first inning but then shut the door the rest of the way, yielding six hits while walking two and striking out four.
The left-hander logged his seventh straight start without a loss, tossing 62 of his 101 pitches for strikes.
David Murphy supplied Texas' offense with a two-run single in the first inning and the American League West-leading Rangers claimed the opener of a three-game series.
San Diego's Jason Marquis (1-2) absorbed the loss despite striking out 10 over seven innings.
Notable deaths of 2014 [PHOTOS]