The World No 14 outclassed Russian Olga Puchkova, with a 6-2 6-2 win reports News.com au.
Puchkova, ranked 85th in the world, struggled throughout the match against Stosur, winning just 11 points from seven service games.
She won two of 18 service points in the first set, winning her only two games with service breaks against the 2011 US Open champion.
Stosur will next face 23rd seed Sabine Lisicki a former Wimbledon semi-finalist known for her powerful serve.
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