The world number three secured a 6-4 6-3 victory in an hour and 44 minutes to move a step closer to winning the WTA International hard-court event on home soil.
Schiavone must have fancied her chances of springing a surprise when she broke Radwanska to take a 4-2 lead in the first set, but the Pole hit straight back and secured two breaks of her own to take a 5-4 advantage before serving out the set.
Radwanska then took charge of the second set by going a break up at 2-1 and Schiavone had to dig deep to save three break points in the fifth game.
Schiavone was unable to deny the impressive Radwanska when she served to stay in the match at 5-3 down though, as the 25-year-old broke again to ease into the last eight.
Fourth seed Alize Cornet was made to work harder for her place in the quarter-finals as she beat qualifier Kristina Kucova 6-3 4-6 6-3.
The French world number 22 was up against it at 3-0 down in the first set, but then reeled off the next six games to take the first set and overcame a second-set blip to progress.
It was plain sailing for Yvonne Meusburger as she beat Mirjana Lucic-Baroni 6-2 6-3.
Croatian Lucic-Baroni struggled with her serve as she double-faulted 10 times. Although she asked a few questions of Meusburger in the second set, the Austrian came through with little trouble.
In the final game of the day, seventh seed Magdalena Rybarikova swept aside Spaniard Silvia Soler-Espinosa 6-2 6-0 in just less than an hour.