Anastasia Pavlyuchenkova beat Sloane Stephens 6-7 (3/7), 6-3, 6-3 to set up a last eight clash with Danielle Collins.
Serena Williams won her 23rd title and seventh at Melbourne Park while eight weeks pregnant two years ago.
Simona Halep takes the top seeding among the women courtesy of her world number one ranking.
To win the decisive tie-break, a player will need to be the first to 10 with an advantage of at least two points.
Roger Federer became the first player to win three Hopman Cups, teaming with Belinda Bencic for the duo's second consecutive title.
World number two Angelique Kerber was too good for the 15th-ranked Barty in the opening match, winning 6-4, 6-4.
The 37-year-old kept his unbeaten record this week intact against the 15th-ranked Stefanos Tsitsipas.
Caroline Wozniacki and Angelique Kerber suffered shock defeats at the hands of less fancied opponents.
World number two Caroline Wozniacki and Wimbledon champion Angelique Kerber eased through their opening matches at the Wuhan Open in China on Tuesday.
Romania's Simona Halep continued to lead Women's Tennis Association (WTA) rankings released on Monday, ahead of Denmark's Caroline Wozniacki.
Roger Federer will be seeking a 13th appearance in the US Open quarter-finals when he plays Australian John Millman.
Sharapova needed just 66 minutes to see off 12th-seeded Kasatkina in an impressive and timely display of big-hitting with the US Open getting underway in just under three weeks' time.
Serena Williams is seeking to regain her top form after missing most of the 2017 WTA season due to a pregnancy.
Serena Williams moved up one spot in the WTA rankings this week to world No.27.
Novak Djokovic won his fourth men's singles title while Angelique Kerber became the first German in 22 years to win the women's crown.
Serena Williams' husband Alexis Ohanian on Saturday posted a heartfelt message on his Instagram account following the tennis star's loss to Angelique Kerber in the final at Wimbledon.
Angelique Kerber became the first German woman to win Wimbledon for 22 years.
Wimbledon Women's Final highlights, Serena Williams vs Angelique Kerber Tennis Match: Angelique Kerber became the first German to win a Wimbledon singles title since Steffi Graf in 1996.