Denmark's Wozniacki fell in an 4-6, 6-4, 6-3 upset to Giorgi in the third-round evening session at New York. Giorgi now goes on to face countrywoman Roberta Vinci following her 6-4, 6-3 decision over 10th-seeded fellow Italian Karin Knapp.
Azarenka, meanwhile, had to overcome a slow start to get past Frenchwoman Alize Cornet 6-7 (2-7), 6-3, 6-2, moving into the round of 16. The Belarusian was last year's U.S. Open runner-up to Serena Williams.
Her next opponent will be 13th-seeded Ana Ivanovic, who rallied from a 4-5 deficit in the second set to down American Christina McHale.
American Alison Riske also moved on with a 6-3, 6-0 surprise win over seventh-seeded Czech Petra Kvitova, the 2011 Wimbledon champion. Kvitova committed 27 unforced errors and seven double-faults in the uninspired effort.
Riske now goes on to face Slovakian Daniela Hantuchova, who notched a 3-6, 7-5, 7-6 (7-4) comeback over to Israel's Julia Glushko.
Also Saturday, 21st-seeded Simona Halep advanced with a 6-1, 6-0 romp over 14th-seeded Russian Maria Kirilenko 4-6, 7-5, 6-4.