Radwanska, the former world number two who is seeking a 15th career title in her native Poland, was rarely worried by Wickmayer and wrapped up a 6-1 6-3 victory to set up a second round meeting with another Belgian, An-Sophie Mestach.
But there was no double family celebration as Radwanska’s younger sister Urszula crashed out 6-0 6-1 to Russia’s Elizaveta Kulichkova.
Fifth seed Kaia Kanepi, of Estonia, overpowered Israeli qualifier Shahar Peer 6-1 6-2 in 53 minutes and Belgian seventh seed Kirsten Flipkens had little trouble in dispatching Germany’s Annika Beck 6-4 6-1.
Anna Karolina Schmiedlova, the eighth seed from Slovakia, dropped a set to Uzbekistani qualifier Nigina Abduraimova but eventually prevailed 6-1 4-6 6-1 after 97 minutes.
Czech pair Klara Koukalova and Katerina Siniakova both progressed with respective 6-2 6-4 and 6-4 5-7 6-2 victories over Petra Martic and Maria-Teresa Torro-Flor.
There were also first-round wins for Polish qualifier Magda Linette and Germany’s Carina Witthoeft.