This is Harrison's fifth major league season and he had not had a shutout in three years. He did not walk a batter and struck out four while throwing 111 pitches.
Texas had lost its previous two games and San Francisco had won two straight.
Kinsler hit a solo homer leading off the first and Josh Hamilton slugged his 22nd of the year in the fifth to give the Rangers a 3-0 lead.
Kinsler singled in a run in the sixth and Craig Gentry had RBI singles in the fourth and ninth among his five hits. Gentry raised his batting average to .341 in 88 at-bats this season.
Barry Zito (5-3) took the loss, giving up four runs on six hits in nine innings.