Wickmayer needed three sets but managed to rally past Russian Valeria Savinykh 4-6, 6-1, 6-4 in a match that lasted just under 2 hours at the Sierra Madre Tennis Club.
Joining her in the second round will be Radwanska, who outlasted American Maria Sanchez 6-2, 2-6, 6-2 on the Mexican hard courts.
Also logging Day 1 wins Monday were Russia's Galina Voskoboeva, Romania's Monica Niculescu and Timea Babos of Hungary.
A match between fifth-seeded Russian Anastasia Pavlyuchenkova and countrywoman Nina Bratchikova was suspended in the first set due to rain.
Notable deaths of 2014 [PHOTOS]
UPI Sports Results for September 19, 2014