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 Nina Bratchikova who beat Frenchwoman Claire Feuerstein 3-6 6-3 7-5.
Li raced into the third round when she brushed aside Frenchwoman Stephanie Foretz Gacon 6-0 6-2.
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 Lauren Davis, who defeated fellow American Christina McHale 6-1 6-3.
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 Varvara Lepchenko after she upset 19th seed and former world number one Jelena Jankovic of Serbia 7-6(4) 4-6 6-4.
Ninth seed Caroline Wozniacki was too good for Australian Jarmila Gajdosova, beating her 6-1 6-4.
Sixteenth seed Maria Kirilenko was dumped out 6-4 3-6 6-3 by Klara Zakopalova.
Dutchwoman Arantxa Rus earned a spot against Julia Goerges after triumphing 6-3 7-6(3) over France’s Virginie Razzano.
German 25th seed Goerges was too good for Briton Heather Watson, winning 6-3 6-4 (see separate report).
Spaniard Carla Suarez Navarro will play Kazakh Yaroslava Shvedova after they secured victories over Sesil Karatantcheva – another Kazakh, by 4-6 6-4 6-1 – and Swede Sofia Arvidsson, 6-1 6-0, respectively.
And 28th seed Peng Shuai beat Spain's Lourdes Dominguez Lino 6-4 6-3.