Ashleigh Barty displaced Japan's Naomi Osaka as women's world number one.
Maria Sharapova's last win over Serena Williams came at the 2004 WTA Tour Championships -- before even the birth of Twitter and YouTube.
Serena Williams, seeking a 24th Grand Slam title, will face Germany's Julia Goerges for a place in the French Open last 16.
Organisers of the Sydney tournament, a warm-up for the first Grand Slam of the year, said Julia Goerges had pulled out with a right knee injury.
The 1-6, 6-3, 6-4 defeat of Caroline Wozniacki, the Danish third seed and former world number one, eliminated the last of the WTA Auckland Classic drawcards, a day after the exit of the two top seeds Serena and Venus Williams.
Julia Goerges cruised past double Grand Slam winner Svetlana Kuznetsova 6-0, 6-1 in the first round of the Dubai Tennis Championships.
Hingis consistently showed the deft hands that made her a nine-time Grand Slam doubles champion. The 32-year-old is scheduled to play doubles with Hantuchova in four tournaments, including the U.S. Open.
Kerber, ranked sixth in the world, broke Kudryavtseva 10 times, but dropped her own serve eight times to allow her Russian opponent to stay in the match.
The sixth seed, a finalist at Melbourne Park in 2011, won 6-4, 6-1 in 71 minutes and will next play 18th-seeded German Julia Goerges, who beat Li's compatriot Zheng Jie in their third round clash.
Zheng was never going to be a pushover. Once ranked 15 in the world with four career singles titles to her name, she has slipped to 40 but had the measure of Stosur in the first set.