Boulter, seeded 31st in the final grand slam of the year, joined compatriot Harriet Dart in the next stage after a 5-7, 6-2, 6-1 win over Aliaksandra Sasnovich.
The win was far more testing than the scoreline suggests as she endured a battle in sweltering New York conditions.
She started brightly, winning the opening two games before the Belarusian started causing problems with a strong forehand weapon. Sasnovich was able to turn the first set around, breaking at 5-5 to assume control.
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