Robson, bidding to become the first British woman in the quarter-finals of a Grand Slam since Jo Durie reached the last eight of Wimbledon, started brightly, earning an early break against the defending champion.
But Stosur grew into the contest bit by bit, and though Robson was more than a match for her in power, the 18-year-old hit too many unforced errors as the first set slipped away.
It was the same story in the second set, though there was late drama as Robson saved eight match points over three games in the closing stages.
"On the match points I probably had a few flashbacks to a few matches last year when I had match points and couldn't close it out," Stosur said.
"But I thought it was a good match and I played pretty well. Towards the end, I probably stepped it up another gear. It was a really good match and definitely one that I needed going to the
Robson produced numerous individual winners, and demonstrated power and groundstrokes that were, at times, too much for the defending champion to live with - just as former Grand Slam champions Kim Clijsters and Li Na had discovered earlier in the week.
But Stosur, realising that Robson's ultra-aggressive approach was leading to mistakes, looked to bide her time, draw Robson into rallies, and wait for the errors.
"I think Laura is a very good player," the Australian said afterwards. "She's very young and she's got a lot of years ahead of her. She hits the ball great and thinks things through. She's one to watch."
Having gone a break down early on, the seventh seed levelled immediately, and once she had fended off a break point opportunity for her young adversary in game nine, she turned the tables by racing to three break and set points on the Robson serve.
Robson denied her on the first two, only to cede the opening set with a double fault.
The task grew more difficult for Robson when she dropped serve again four games into the second set - once more, having missed out on break points in the previous game.
Stosur looked to be moving comfortably towards a quarter-final showdown with top seed Victoria Azarenka, the Belarusian beating Anna Tatishvili 6-2 6-2, only to spurn five match points in a 15-minute epic game at 5-2 on the Robson serve.
Robson showed character to come through, and once more to overturn a 40-15 deficit on the Stosur serve to break back afresh.
In the end, however, Robson gave her opponent one chance too many, but she will take heart from what has been a breakthrough tournament for her, and one that will catapult her from a world ranking of 89 to a new high when the next set of standings are announced at the end of the US Open.