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After battling through qualifying, Garcia beat three top-10 world-ranked opponents, Maria Sakkari, Aryna Sabalenka and Jessica Pegula, on the way to the final where she delivered a master class against Czech Kvitova.
Petra Kvitova overcame a sluggish start and 10 double faults to beat Madison Keys 6-7(6), 6-4, 6-3 and advance to the women's singles final, where she will face Caroline Garcia, the first qualifier to reach the final of a WTA 1000 tournament.
In the men's section, world No. 1 Daniil Medvedev and fourth seed Stefanos Tsitsipas defused big-serving American threats Taylor Fritz and John Isner to reach the semifinals.
Second seed Anett Kontaveit also had her preparations for Flushing Meadows cut short in the round of 16.
Kasatkina thundered past Italy's Giorgi 6-2, 6-2 with a near flawless baseline game.
The top seed Barty defeated Madison Keys 6-1, 6-3 in the semifinals.
The 25-year-old reached the semifinals at Melbourne Park for the second time in three years after blitzing American Jessica Pegula 6-2, 6-0 in just over an hour.
Keys booked her first Grand Slam semifinals in four years with a 6-3, 6-2 upset of Krejcikova.
Results 1-8
