Reigning U.S. Open champ Stosur, who was the French runner-up in 2010, handled Elena Baltacha of Britain 6-4, 6-0 on the red clay courts at Roland Garros in Paris.
Meanwhile, 2009 champion Kuznetsova breezed past Croatia's Mirjana Lucic 6-1, 6-3.
Also notching a win Sunday was unseeded American great Venus Williams, who rallied to defeat Argentina's Paula Ormaechea 4-6, 6-1, 6-3.
Sunday's other winners included 10th-seeded German Angelique Kerber, No. 20 Lucie Safarova of the Czech Republic, Spaniard Maria Jose Martinez Sanchez, Italian Sara Errani, Americans Melanie Oudin and Irina Falconi, Irena Pavlovic of France and Aleksandra Wozniak of Canada.