Wozniacki, the seventh seed and World number 10, was beaten 7-6 (7/6) 6-1 Carla Suarez Navarro following a marathon first set which lasted 70 minutes after Wozniacki had taken a 3-0 lead.
Suarez Navarro came back to win on a tie break before she cruised through the second set to book her place in the next round against Yaroslava Shvedova.
Stosur, seeded sixth, also failed to reach the second round after she was beaten 6-4 6-4 by Jelena Jankovic.
Meanwhile, fifth seed Petra Kvitova survived a scare of her own against Annika Beck to overturn an early 4-1 deficit to take the first set before she lost the second and fought back in the third to win 7-5 6-7(0/7) 6-3.
Anastasia Pavlyuchenkova won a two hour 37 minute match against Ekaterina Makarova, 6-7(5/7) 7-5 6-4 and Yaroslava Shvedova defeat Roberta Vinci 6-4 6-2.
Lucie Safarova, Shvedova and Julia Goerges all came through their first round matches while qualifier Bethanie Mattek-Sands of the United States defeated Germany's Yanina Wickmayer 6-2 6-1.