Kvitova made a break in the first game of the match stand for a set win. Kvitova also opened the second set with a break but Radwanska immediately answered only to have Kvitova break again for a 2-1 lead. Radwanska gave herself four break chances in the 10th game but Kvitova held on, ending play on her fourth match point.
Williams, the world No. 1, pasted eighth-seeded Angelique Kerber 6-3, 6-1 in 66 minutes. She won a healthy 76 percent of the points on serve and saved both of Kerber's break chances.
Williams has assured herself of a $10 million year -- a figure that could go to $12 million if she repeats as WTA Championships winner. It is record for a WTA earnings year. The previous mark was just less than $8 million by Azarenka last year.
Azarenka, No. 2 in the world rankings, began her tournament by toughing out a 7-6 (7-4), 6-2 battle with sixth-seeded Sara Errani. Azarenka had to overcome dropping serve five times in 10 games and she won barely half (43-of-79) of the points on serve.
She goes against seventh-seeded Jelena Jankovic in a White Group match Wednesday. Errani meets fourth-seeded Li Na in another White Group match. Radwanska meets Williams, against whom she is 0-7, in a must-win situation in the Green Group.
The top two players from each group's round-robin play advance to the TEB BNP Paribas WTA Championships semifinals Saturday.