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Azarenka clinched a 6-1, 6-4 win over Elena Vesnina - her fifth win in five matches against the experienced Russian - and will next face German 19-year-old Annika Beck, the 2012 junior champion, for a place in the third round.
Roland Garros has been Azarenka's least successful Grand Slam.
Having reached at least the semi-finals of the other three majors, the 23-year-old's best run in Paris saw her make the last eight in 2009 and 2011.
But she was relatively untroubled on Wednesday against Vesnina, with whom she reached the final of the women's doubles in 2009.
Breaks in the second and sixth games of the opening set gave her the foundation for victory in a match held over from Tuesday because of rain.
She was 2-1 up with a break in the second before the 28-year-old Vesnina rallied to level at 3-3.
But it was a brief respite for the world number 38 as Azarenka hit back straightaway to seal the tie after 78 minutes.
"It felt good to be on the court after getting cancelled yesterday," said Azarenka.
"It was a good win as Elena is a really tough opponent.
"I will be back tomorrow to play again but I want to be able to slide better on the clay, especially on the left foot as well as the right. That's my goal this year."
Polish fourth seed Radwanksa, who has never got beyond the fourth round, eased past Mallory Burdette of the United States 6-3, 6-2.
She could next face sister Urszula for a place in the last 16 if the younger sibling, who put out Venus Williams in her opener, beats German qualifier Dinah Pfizenmaier.
