Seventh-seeded Świątek beat countrywoman Magdalena Frech 6-2, 6-1, while No. 16 seed Chwalińska rallied past Czech player Marie Bouzková 1-6, 6-1, 6-1.
Świątek, ranked seventh in the world, will face American Ashlyn Krueger, ranked 68th, in the third round.
Chwalińska's next opponent will be 10th-seeded American Iva Jovic.
Chwalińska's victory against Bouzková moved her to a career-high No. 20 in the live rankings, Poland's PAP news agency reported.
The 24-year-old had previously reached No. 21.
The Wuhan Open runs until Sunday and is one of the 10 WTA 1000-level tournaments on the Women's Tennis Association calendar, one tier below a Grand Slam.
(gs)
Source: IAR, PAP