Świątek, a former world No. 1 now ranked eighth in the world, beat Podoroska 6-3, 6-2 in their second-round match at Flushing Meadows.
She will next face the Czech Republic's Marie Bouzková at the last Grand Slam tournament of the year.
Świątek, the 2022 US Open champion, is seeking her seventh Grand Slam singles title in New York.
The 25-year-old is the last Polish singles player remaining in the tournament, after countrywomen Maja Chwalińska, Magdalena Fręch and Magda Linette all crashed out in the first round.
On the men's side, Poland's Hubert Hurkacz and Kamil Majchrzak were both eliminated in the second round.
The 2026 US Open runs until September 13.
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Source: Eurosport, IAR/PAP