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Caroline Wozniacki also advanced on Wednesday after the third-seeded Danish player opened her title defence by beating Romania's Simona Halep 6-2, 6-3 in 1 hour, 11 minutes.
The fourth-seeded Stosur of Australia broke twice in both the first and third sets, and saved all nine break points she faced in the match.
Earlier on Wednesday, top-ranked Victoria Azarenka pulled out of the Dubai tournament because of a left ankle injury. The Australian Open champion, who has won 17 straight matches so far this season, says she will rest for two or three days.
"Obviously it's very disappointing, but, I mean, it's life. It's sport. I just have to take care of it and that's it," Azarenka said. "I take a couple of days off just to see and hopefully it will settle down... but it has to be taken care of right away."
Azarenka injured her left ankle during last week's semi-final match at the Qatar Open. However, Azarenka shook off the injury to beat Stosur in the final.
In Dubai, Stosur will meet eight-seeded Jelena Jankovic of Serbia in the third round on Thursday. Jankovic defeated Italy's Flavia Pennetta 7-5, 6-4.
Wozniacki will face Ana Ivanovic after the Serbian player defeated Maria Kirilenko of Russia, 6-2, 7-6 (7/4).
The ninth-seeded Sabine Lisicki of Germany also advanced after defeating Iveta Benesova of the Czech Republic 6-3, 6-3. Lisicki will play Agnieszka Radwanska in the third round, after the fifth-seeded Polish player defeated Israel's Shahar Peer 7-5, 6-4.
Azarenka said the packed tennis schedule and increasingly tough competition could be partly to blame for so many injuries among the players.
Wimbledon champion Petra Kvitova, French Open winner Li Na and Vera Zvonareva have also pulled out the Dubai tournament because of injury and illness.
"For me, obviously I played so much, and it was an unfortunate situation what happened in Doha," Azarenka said. "But I mean, it's just the game became so physical. Sometimes it's really tough on the body week after week to play. For example, these tournaments back-to-back with such a tough field, it takes a lot out of you physically. It's just tough."
Results from the third day of the Dubai WTA Open on Wednesday (x denotes seeding):
Second round
Caroline Wozniacki (DEN x3) bt Simona Halep (ROM) 6-2, 6-3
Julia Goerges (GER) bt Casey Dellacqua (AUS) 6-0, 6-2
Daniela Hantuchova (SVK) bt Peng Shuai (CHN) 7-5, 6-2
Ana Ivanovic (SRB) bt Maria Kirilenko (RUS) 6-2, 7-6 (7/4)
Jelena Jankovic (SRB x8) bt Flavia Pennetta (ITA) 6-4, 6-2
Samantha Stosur (AUS x4) bt Lucie Safarova (CZE) 6-1, 6-7 (5/7), 6-1
Agnieszka Radwanska (POL x5) bt Shahar Peer (ISR) 7-5, 6-4
Sabine Lisicki (GER x9) bt Iveta Benesova (CZE) 6-3, 6-3
Note - Dellacqua was lucky loser inserted after No 1 seed Victoria Azarenka pulled out injured with a damaged ankle.
