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.
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.
Recent top-10 player Andrea Petkovic of Germany has withdrawn from Wimbledon because of a persistent right ankle injury.
Andy Murray was the last Briton standing after a humiliating first day for the nation at the Australian Open on Monday when five players failed to win a single set.
China's French Open champion Li Na negotiated a tough first match at the Australian Open Monday while third seed Victoria Azarenka laid down an early marker at the year's first grand slam.