Linette, the world No. 80, on Thursday beat Czech player Karolína Muchová 6-7, 6-4, 7-6 in a tough match lasting almost three hours.
Next up for the 27-year-old Pole is Czech number 5 seed Kateřina Siniaková. They face off on Friday evening.
The New York Open is a final chance for players to test their form before the Grand Slam US Open starts on Monday.
(pk)