Williams, the 2007 Wimbledon champion, trailed, 3-4, in the third set but won the next three games and took the match, 7-6, 0-6, 6-4.
Sunday's reverse singles will send six-time major titlist Williams against Chakvetadze and King vs. Petrova. The doubles calls for Americans King and Lisa Raymond to meet Russians Petrova and Elena Vesnina.
The United States-Russia winner will meet the France-Italy winner in the World Group final in mid-September.
Francesca Schiavone beat former world No. 1 Amelie Mauresmo in straight sets Saturday to help Italy draw even with France. Schiavone won, 7-5, 6-3.
Earlier, France's Tatiana Golovin defeated Italy's Tathiana Garbin, 6-3, 2-6, 6-2.