Kvitova, the tournament's fifth seed from the Czech Republic, defeated France's Stephanie Cohen-Aloro 6-4, 6-0, while seventh-seeded King beat Germany's Kathrin Woerle 6-3, 6-4.
The 21-year-old King, No. 73 in the world rankings, was successful on 61 percent of her first serves in the 90-minute match with Woerle, ranked 124th.
Kvitova, No. 68, converted 60 percent of her first serves, including a handful of aces. The 19-year-old Kvitova won 92 percent of her first serve points.