On a day in which results went mostly according to plan at the clay-court tournament, Anastasia Pavlyuchenkova also made it to the third round, Venus Williams continued her comeback from illness with a first-round victory and No.10 seed Anabel Medina-Garrigues was the highest-ranked casualty.
Australia's Stosur, seeded fifth and the highest-ranked player left in the tournament after Agnieszka Radwanska's withdrawal, beat American Jamie Hampton 6-0, 7-5.
No.4 Zvonareva of Russia cruised past Ukraine's Kateryna Bondarenko 6-1, 6-1, and Serena Williams was a 6-3, 6-4 winner over Russia's Elena Vesnina.
Serena, playing this tournament for the first time since she took the title four years ago, had 11 aces in her first match on clay in more than a year.
"I forgot I hadn't played on clay in so long," she said. "I feel like it really suits my game. I don't have to go crazy and move my feet so much. I pretty much just go for my serve and do the best I can."
Sister Venus, who is making her way back from an autoimmune disease, had a 6-4, 6-3 win over Czech Iveta Benesova, setting up a second-round clash with Jelena Jankovic.
"This is great for me to be in another match and to be in the second round," she said. "It was a good start."
The elder Williams returned to the tour last week after a seven-month break, winning four matches at the Sony Ericsson Open in Key Biscayne before eventually losing to Radwanska, who went on to win the title.
Venus Williams, who has 43 career titles, said she takes a lot of ice baths and had three days off just to rest following Thursday's loss.
"For me it's just about having energy at this point. I try not to worry about tomorrow," she said.
"There's so many opportunities going forward and I try to focus on the forward and focus on every day and today's win takes me closer."
Williams has won seven Grand Slam tournaments but is now ranked No. 87. She won the Family Circle Cup eight years ago and is unseeded this time.
No. 8 seed Pavlyuchenkova ousted Eleni Daniilidou 3-6, 6-4, 6-4 while No.10 Medina-Garrigues was beaten 7-5, 6-3 by Australia's Anastasia Rodionova.
No. 12 seed Yanina Wickmayer of Belgium defeated Vania King of the U.S. 6-4, 6-3 while No. 13 Nadia Petrova beat Mariana Duque-Marino 6-1, 6-2, and No.17 Marina Erakovic of New Zealand advanced with a 6-2, 6-4 defeat of Karolina Pliskova.