The 15-year-old Coco Gauff reached her first-ever WTA final and became the youngest woman to do so in 15 years.
Andrea Petkovic eased past Italian Camila Giorgi in her opening game at the Dubai Open.
Jiri Vesely defeated Lleyton Hewitt, who will retire after the Australian Open, 6-3, 6-7 (1), 6-4 before Karolina Pliskova came from 3-1 down in the second set to beat Jarmila Wolfe Gajdosova 7-5, 6-3 at Perth, Western Australia.
Footage of the incident quickly did the rounds on social media, with some contributors accusing her of racism. But on Monday, Petkovic insisted she didn't mean to be rude.
Russia's Kuznetsova, the winner of 2009 French Open and 2004 US Open, lost the lop-sided match 0-6, 1-6 to her compatriot Anastasia Pavlyuchenkova.
Germany, the 2014 finalists, will face Russia away in the semi-finals on April 18-19 after Petkovic again dug deep to seal a 6-3, 3-6, 8-6 win over Jarmila Gajdosova, with the third set of their fourth singles rubber taking nearly an hour.
The Czechs are eyeing their third Fed Cup win in four years after lifting the trophy in 2011 and 2012. Germany are looking for their first title since 1992, when a team led by Steffi Graf and current captain Barbara Rittner won the title.
Monica Puig, seeded eighth in the event and ranked 48th in the world, outlasted 92nd-placed Watson 6-4, 6-4 in 94 minutes.
Sania and Huber will now fight it out with Romania's Monica Niculescu and Czech Republic's Klara Zakopalova for a place in the semifinals.
Laura Robson was sent packing Monday by former world number one Caroline Wozniacki and Elena Baltacha won just three games Tuesday in losing to New Zealand's Marina Erakovic.
World No.2 Maria Sharapova is aiming to become just the third player, after Steffi Graf and Kim Clijsters to win back-to-back Indian Wells and Sony Open titles.
Poland's Radwanska is on an unmatched win run this year after claiming tournaments in Auckland and Sydney, and next faces one of two former world number ones, Ana Ivanovic or Jelana Jankovic.
Eighteen-year-old Elina Svitolina of Ukraine denied veteran Kimiko Date-Krumm a date with history when she overpowered the Japanese 6-2 6-3 to lift the Indian Open, her maiden WTA title.
Comeback German girl Andrea Petkovic toppled top seed Nina Bratchikova of Russia while Japanese veteran Kimiko Date-Krumm shocked compatriot and second seed Misaki Doi to storm into the semifinals of the WTA Indian Open, on Friday.
Having waited 76 years for a men's Grand Slam champion and 24 years to celebrate a women's tour winner, Britain, once the cash-heavy, running joke of world tennis, is suddenly a serious player.
Maria Sharapova came back from one set down to beat Heather Watson 6-7 (9/7), 6-3, 6-4 in the second round of the Pan Pacific Open in Tokyo on Tuesday.
Polish third seed Agnieszka Radwanska reached the Wimbledon last 16 on Friday with a 6-0, 6-2 win over Britain's Heather Watson.
Heather Watson ended Britain's 10-year wait to have a woman in the third round at Wimbledon as the youngster swept to a 6-1, 6-4 victory over Jamie Hampton of the United States on Wednesday.