Stephens needed three sets and nearly three hours to win her opening match. But this one was far easier, as she hit 24 winners and only five unforced errors.
Radwanska was joined in the quarter-finals by younger sister Urszula, who stunned sixth-seeded Serbian Jelena Jankovic 6-2, 4-6, 6-3
Radwanska, ranked fourth in the world and seeking her third title of 2013 after triumphs in Auckland and Sydney, had to battle to subdue her 40th-ranked opponent in two hours and 14 minutes.
The 33-year-old - who has won seven Grand Slam titles in all - also pulled out of the doubles there with her sister Serena because of a back problem.
In other second-round matches, fourth-seeded Tamira Paszek of Austria lost to American Alison Riske, 2-6, 6-2, 6-3, and eighth-seeded Urszula Radwanska of Poland was beaten by Croatia's Donna Vekic, 6-3, 6-2.
Urszula Radwanska has never made it past the second round in 13 appearances at Grand Slam tournaments.
Filling a mostly empty stadium court with her familiar shrieks on Wednesday, the former world No. 1 beat Elena Vesnina of Russia 6-1, 6-4.
The quick exit for the 32-year-old Williams came a year after her younger sister Serena was upset in the first round at Roland Garros.
Azarenka was replaced in the Miami draw by 19-year-old US lucky loser Lauren Davis, who saved three match points in the deciding tiebreaker to defeat compatriot Madison Keys 6-1, 5-7, 7-6 (9/7).
Jelena Jankovic overcame problems with her serve to claim a 2-6, 6-4, 6-4 win over Melinda Czink in the second round of the Mercury Insurance Open on Thursday.
Poland's Urszula Radwanska advanced to the final of the s-Hertogenbosch grasscourt tournament on Friday after Belgian Kim Clijsters pulled out of their semi-final clash citing injury.
Czech fourth seed Petra Kvitova enjoyed a second comfortable outing at the French Open on Thursday, reaching the third round with a 6-1, 6-3 win over Poland's Urszula Radwanska.
Rising Serbian teen Bojana Jovanovski overcame Urszula Radwanska of Poland 1-6, 7-5, 6-4 to reach the Malaysian Open quarterfinals on Wednesday.