The second seed from Russia had too much poise and pace for Stosur, runner-up in 2010 and twice a semi-finalist, whose heavy top-spin game can be a handful on clay.
Sharapova, who played in the last three French Open finals, winning two, broke four times to seal a convincing win when Stosur fired a forehand wide.
The five-time Grand Slam champion will take on Czech 13th seed Lucie Safarova for a quarter-final spot.
Elsewhere, Alize Cornet needed two and a half hours to defeat 33-year-old Croatian Mirjana Lucic-Baroni 4-6 6-3 7-5 in a match scarred by 85 unforced errors and a bizarre incident where the French player served to the wrong side.
Elina Svitolina, the 19th seeded Ukrainian, reached the fourth round of a major for the first time – she defeated fellow former French Open junior champion Annika Beck of Germany 6-3 2-6 6-4.