Azarenka of Belarus cruised to a 6-0, 6-1 thrashing of Russia's Alexandra Panova while No. 7 seed Stosur needed only 51 minutes to dismantle Petra Martic of Croatia 6-1, 6-1 at Arthur Ashe Stadium.
Stosur, 28, bested Serena Williams in last year's New York finale to claim her lone career Grand Slam title.
Third-seeded Russian Maria Sharapova, meanwhile, humbled Hungarian Melinda Czink 6-2, 6-2 in another first-round matchup, while 23rd-seeded Kim Clijsters notched her 22nd straight U.S. Open win with a 6-3, 6-1 victory over 16-year-old American Victoria Duval.
Clijsters -- who won the Grand Slam event in 2005, 2009 and 2010 -- is playing in what she has said is the final tournament of her career.
Petra Kvitova of the Czech Republic, coming off a title last week in New Haven, posted a 7-6 (8-6), 6-1 victory over Slovenia's Polona Hercog.
Also Monday, China's Li Na, who claimed the Western & Southern Open last week, waited out a long rain delay to advance with a 6-2, 6-3 victory over Britain's Heather Watson.
No. 11 Marion Bartoli of France bested American Jamie Hampton 6-3, 7-6 (7-5), No. 15 Czech Lucie Safarova dispatched American Melanie Oudin 6-4, 6-0 and 19th-seeded Nadia Petrova of Russia downed Aussie Jarmila Gajdosova 6-3, 7-6 (7-3) in other action.
Also emerging with Day 1 wins were Russians Anastasia Pavlyuchenkova and Nadia Petrova, Belgian Yanina Wickmayer, American Varvara Lepchenko and China's Zheng Jie.
Logging opening day victories were Romanian Simona Halep, Aussie Casey Dellacqua, Georgian Anna Tatishvili, Luxembourg's Mandy Minella, Swiss Romina Oprandi, Spaniard Lourdes Dominguez Lino, American Mallory Burdette, Frenchwoman Pauline Parmentier, Romanian Edina Gallovits-Hall, Australian Anastasia Rodionova, Britain's Laura Robson, France's Alize Cornet, Belgian Kirsten Flipkens, Slovakian Magdalena Rybarikova and Canadian Aleksandra Wozniak.