The 17-year-old Duval stunned the USTA Billie Jean King National Tennis Center crowd with a 5-7, 6-4, 6-4 comeback victory over the 2011 champion while playing in just her second grand slam match.
Stosur gave her plenty of help, committing 56 unforced errors and 10 double faults, finally losing on Duval's fourth match point.
Azarenka, meanwhile, had little trouble reaching the second round, rolling over Germany's Dinah Pfizenmaier 6-0, 6-0 at Arthur Ashe Stadium.
The Belarusian was runner-up in last year's U.S. Open to current world No. 1 Serena Williams.
Italy's Errani made quick work of Australian Olivia Rogowska, dispatching her 6-0, 6-0 under warm and breezy conditions in New York, while Denmark's Wozniacki also recorded a straight-set decision, 6-2, 7-5 over Chinese Duan Ying-Ying.
Joining them in the second round were seventh-seeded Petra Kvitova of the Czech Republic, who doused Japan's Misaki Doi 6-2, 3-6, 6-1 in a 1-hour, 55-minute match, and 10th-seeded Italian Roberta Vinci, who dismissed Hungarian Timea Babos 6-4, 6-2.
Among the Day 2 winners were No. 21 Simona Halep of Romania, Ukrainian Elina Svitolina, Israeli Julia Glushko, No. 22 Russian Elena Vesnina, No. 26 Alize Cornet of France, No. 27 Svetlana Kuznetsova of Russia, No. 28 Mona Barthel of Germany, Taiwan's Su-Wei Hsieh, and Americans Christina McHale and Sachia Vickery.