Azarenka gained the Round of 8 with a 6-2, 6-2 thrashing of Georgian Anna Tatishvili. The world No. 1 from Belarus, who has lost just 10 games in New York, gained her first U.S. Open quarterfinal.
She will next meet the seventh-seeded Stosur, who dispatched 18-year-old Briton Laura Robson 6-4, 6-4. The Australian reeled off six aces against Robson, who had earlier bested heavily favored Kim Clijsters and Li Na.
Third-seeded Russian Maria Sharapova also won her fourth-round match in a rain-delayed night contest, rolling over Nadia Petrova 6-1, 4-6, 6-4.
The French Open winner will now take on 11th-seeded Frenchwoman Marion Bartoli, who rallied for a 1-6, 6-2, 6-0 triumph over No. 5 Petra Kvitova of the Czech Republic.
Kvitova lost for only the second time 17 matches.
2014: The Year in Music [PHOTOS]