Harrison (10-8) yielded five hits over seven innings to earn the victory, striking out six while issuing no walks as the Rangers took the opener of a three-game series.
Texas banged out 15 to back Harrison up, led by a 3-for-4 performance from Nelson Cruz, who smacked a homer and drove in a pair of runs.
Yorvit Torrealba and Mitch Moreland added three hits apiece for the American League West leaders, who won for the fourth time in five games.
Charlie Furbush (2-4) was tagged with the loss, surrendering seven runs (six earned) on eight hits before exiting after four innings.
Jack Wilson hit a two-run double for the Mariners.