Australia's Samantha Stosur, the No. 5 seed, beat Carla Suarez Navarro, from Spain, 6-1 6-3.
Williams will next play 32nd seed Carla Suarez Navarro, who beat Germany's Andrea Petkovic 6-1 6-4.
In the women's final, Italian top seed Sara Errani breezed past second seed Carla Suarez Navarro of Spain 6-0 6-4.
Wozniacki, seeded 10th, beat Ukraine's Lesia Tsurenko 6-4 6-3 while Kuznetsova overcame Spain's Carla Suarez Navarro in three sets 6-2 4-6 6-3.
In other action Sunday, former Wimbledon champion Petra Kvitova made an impressive start with a 6-3 6-4 win over Spaniard Carla Suarez Navarro.
Russia later made in 4-0 as Ekaterina Makarova and Elena Vesnina defeated Nuria Llagostera Vives and Carla Suarez Navarro 6-2 6-1 in the doubles.
The French 32nd seed came from behind to win 2-6 6-3 6-2 and earn a clash with either Italian Roberta Vinci or Spain's Carla Suarez Navarro.
The 21-year-old Czech, who can usurp Caroline Wozniacki at the top of the world rankings in the next fortnight, faces Spain's Carla Suarez Navarro in round two.
The World No.1 will face Spain's Carla Suarez Navarro in the semifinal, but says she is taking nothing for granted after her shock defeat in the opening round at Roland Garros last year.
The women's matchups Sunday to decide quarterfinal berths are: 15-time major champion Serena Williams against No. 15 Roberta Vinci, 2008 French Open titlist Ana Ivanovic against No. 4 Agnieszka Radwanska, 2009 champion Svetlana Kuznetsova against No. 8 Angelique Kerber, and 2012 runner-up Sara Errani against No. 20 Carla Suarez Navarro.
But Serbian pair Ana Ivanovic and Jelena Jankovic were third round casualties, losing to Romina Oprandi and Carla Suarez-Navarro respectively.
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