The opening two sets were split between the pair before Azarenka knocked the defending champion out of Flushing Meadows by winning a tie-break in the final set.
"She never gives up," Stosur told reporters. "She fights very hard every single match she plays and she does it very well.
"That's probably part of the reason she's number one at the moment. She's one of the players you have to get out there and she barely gives you any points and you have to work hard for every single point you get."
The duo had never met at the quarter-final stage of Grand Slam prior to today.