Stosur claimed the crown two years ago and, after seeing off Ying-Ying Duan in her first-round match, had few problems in disposing of Madison Brengle 6-0 6-1.
Her last-eight opponent will be fourth seed Suarez Navarro, who overcame Christina McHale 6-4 6-1.
Kristyna Pliskova and Shuai Peng also sealed their quarter-final berths thanks to respective wins over Monica Puig and Amandine Hesse.
Meanwhile, at the Nurnberger Versicherungscup, defending champion Kiki Bertens got her 2017 campaign off to a flying start with a 6-2 6-1 romp against Katharina Gerlach.
She will now meet Annika Beck in the second round, after the German saw off compatriot Lena Rueffer 6-3 6-3.
Alison Riske and Sorana Cirstea were the first women to seal their quarter-final places after wins over Kirsten Flipkens and Varvara Lepchenko respectively, while Carina Witthoeft, Yaroslava Shvedova, Laura Siegemund and Yulia Putintseva were winners in the first round.