The sixth-seeded Russian breezed through the first set before the Japanese player fought back.
Kuznetsova, who beat compatriot Elena Dementieva in the 2004 U.S. Open final, regained her composure to wrap up the third set.
Serbia and Montenegro's Ana Ivanovic also moved through with a 6-2, 6-4 win over China's Shuai Peng and will now meet second seed Amelie Mauresmo.
Russia's Maria Kirilenko advanced when her Argentine opponent Gisela Dulko retired in the second set with an abdominal strain.
Kirilenko, who made the quarter-finals in Auckland earlier this week, was leading 6-4, 3-0 when Dulko was forced out.
