Stosur opened with intent against Slovakian Dominika Cibulkova, racing out to an early lead as she broke her opponent twice to begin the first set.
Stosur missed a good chance to break again as Cibulkova hit back to make it 4-3 but the 30-year-old Australian was good enough to hold her last two service games to take the first set 6-4.
And, having achieved an early break in the second, Stosur held serve to complete a 6-4 6-4 victory.
But as Stosur advanced to the last 16, other big names were falling around her as the 2014 French Open continues to prove periolous to the women's seeds.
Never before, though, in the Open era have the three top-seeded women been eliminated before the fourth round of a grand slam.
However, after Li Na and Serena Williams respectively bit the dust in the first and second rounds, Radwanska followed suit in the third with a 6-4 6-4 defeat to Ajla Tomljanovic, completing a wipeout of the leading trio in the draw.
Radwanska lost five consecutive games in the opening set at Roland Garros and, although she rallied late on, her Croatian opponent - ranked 69 places lower - was able to take the lead thanks to some deft drop shots.
Another break at the start of the second set was enough for Tomljanovic to secure a memorable victory as Radwanska's game totally deserted her on Court Philippe Chatrier.
Tomljanovic will go up against Carla Suarez Navarro in the next round, after the Spaniard triumphed 6-2 6-2 over Taylor Townsend, while Romania's Simona Halep is now the top remaining seed in the draw.
As those in the higher seeding positions fall by the wayside, 2012 champion Maria Sharapova is beginning to gather momentum.
The Russian, looking to atone for her final defeat to Williams last year, took just 51 minutes to defeat Paula Ormaechea 6-0 6-0.
Sharapova hit 23 winners and won 88 per cent of her first-service points in a completely one-sided affair.
Ormaechea struggled to cope with the power of the seventh seed, the 21-year-old failing to win a single point on her second serve.
After a shaky start from the underdog, there was to be no let up as Sharapova ruthlessly dispatched almost every opportunity that came her way, and she will now face Stosur.
Garbine Muguruza, who accounted for Serena Williams, continued her impressive form with a 6-2 6-4 victory over Anna Karolina Schmiedlova, setting up a fourth-round tie with French wildcard Pauline Parmentier, who beat Germany's Mona Barthel 1-6 6-1 7-5.
Eugenie Bouchard and Angelique Kerber will also meet in the next round after wins over Johanna Larsson and Daniela Hantuchova respectively.