Kvitova, who had only dropped three games in her opening round, was never threatened by the world number 79, hitting 30 winners in a one-sided match on Court Suzanne Lenglen.
The Wimbledon champion ended Radwanska's misery after 70 minutes on her second match point when the Pole sent a backhand long.
Kvitova, heading towards a possible quarter-final clash with defending champion Li Na of China, will next face Russian
Chinese seventh seed Li, who last year became the first Asian player to win a Grand Slam singles title, needed just 52 minutes to advance on Court One.
Foretz Gacon, the world number 86, won the first two points but that was as close as she came to victory as Li bagged the opening 10 games.
Foretz Gacon pulled a break back but the revival was short-lived with Li setting up a meeting against
Schiavone, the Roland Garros winner in 2010 and runner-up last year, took some time to get her serve in full working order on Court One.
She gave Pironkova, ranked 41 in the world, few openings in the second and third sets, however, and needed only one matchpoint, which she took with a forehand winner, to claim victory.
The 31-year-old Italian, seeded 14th, will now play American
The Dane, a former world number one, will play Estonian
The Czech goes up against 22nd seed
German 25th seed
And 28th seed