Kvitova easily dispensed with France's Stephanie Cohen-Aloro 6-4, 6-0, in a match that took less than an hour on the indoor hardcourts, while King bested German Kathrin Woerle 6-3, 6-4 at The Racquet Club of Memphis.
Also notching a win was No. 7 seed American Vania King, who breezed past German Kathrin Woerle 6-3, 6-4.
Sunday's only other first-round match saw American wild card Bethanie Mattek-Sands put on an impressive display in besting Briton Katie O'Brien 6-2, 6-1.
Celebrity Couples of 2014 [PHOTOS]