Stosur struggled in the first set and lost two of her service games to Morita but in the next two sets Stosur pushed for four breaks of serve, winning five in all out of 11 attempts.
The upset of the day began when Galina Voskoboeva took out ninth-seeded Marion Bartoli in straight sets.
The French Open semifinalist fell 6-3, 6-3 to the Kazakh World No. 135, improving her record to 6-8 over the past 12 months.
This was the qualifier's first win over the French player in their past five meetings.
Bartoli hit four double-faults and was unable to save four of six break points. Voskoboeva won 34 of 48 total service points and 58 percent of the total points overall.
Halep had zero aces and five double faults but was able to win five of seven opportunities on the former No. 1's serve.