The Belarusian star overcame a rain delay to bounce her seventh-seeded Australian opponent 6-1, 4-6, 7-6 (7-5) at Arthur Ashe Stadium in a quarterfinal matchup.
Azarenka was leading 3-0 in the first set when the rains came in New York, delaying the contest for 75 minutes.
After Stosur rallied in the second set, Azarenka jumped out to a 4-2 lead in the final stanza. Stosur got the break and eventually forced a match-deciding tiebreak, but sent a backhand beyond the baseline on the first match point, handing Azarenka the winner.
The 23-year-old world No. 1 improved to 7-0 lifetime overall versus Stosur, and now moves on to her fourth career Grand Slam semifinal and her first one at the U.S. Open.
Bartoli was leading 4-0 in the first set of that match when rain again suspended play.