With the score tied 1-1 and none out in the sixth, Adrian Beltre walked and David Murphy singled off Steve Delabar (1-1). One out later, Napoli crushed a pitch 401 feet off the facing of the second deck to put Texas ahead.
Harrison (6-3) handled the Mariners on seven hits over eight innings, surrendering a pair of runs while walking none and striking out five.
Joe Nathan closed out the victory with his 11th save for the American League West-leading Rangers, who won their fourth straight in the opener of a three-game series.
Mariners' starter Kevin Millwood gave up only one run on four hits in his five-inning stint.
Jesus Montero and Brendan Ryan each drove in a run for Seattle.