In her first appearance in the Tokyo tournament since 2009, Williams took advantage of six-double faults by Azarenka at Ariake Colosseum.
Williams next faces No. 13-seeded Simona Halep, who defeated Andrea Petkovic 7-6 (4), 6-3.
In other second-round matches, second-seeded Agnieszka Radwanska defeated Aleksandra Wozniak 7-6 (5), 6-0. Fourth-seeded Caroline Wozniacki overpowered Flavia Pennetta 6-2, 6-3.
Kuznetsova will next face 15th-seeded Sorana Cirstea, who beat Misaki Doi 6-1, 6-2.
Eugenie Bouchard defeated ninth-seeded American Sloane Stephens 5-7, 7-6 (7), 6-3. She will play sixth-seeded Jelena Jankovic, who beat Ayumi Morita 6-4, 6-1.