The Brit exchanged early breaks with her opponent inside Arthur Ashe Stadium before squandering two break points on the Stosur serve.
The Australian, who looked solid throughout, then broke for a second time to take the opening set 6-4.
Robson wasted three more break-point opportunities at the start of the second before she dropped serve to give Stosur a 3-1 lead.
With Stosur serving for the match at 5-3, the 18-year-old then saved five match points before breaking back as the Aussie failed to close out the match.
However, a final break for Stosur saw her seal the win 6-4, but not before Robson had saved three more match points.