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.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Millions of Getty images now available for free via embed tool