Azarenka, ranked No. 1, got past Aleksandra Wozniak 6-4, 6-4 in a third-round match. Azarenka had seven double faults and 25 unforced errors but only lost her serve twice while recording a pair of breaks in each set.
Sharapova, No. 2, completed the second round with a match that was postponed Thursday. She defeated Ayumi Morita 6-1, 6-1, Morita managed just one winner and didn't force a break-point situation on Sharapova.
Sharapova finished off five of her seven break chances and won 78 percent of the points on serve and 58 when receiving.
Svetlana Kuznetsova, the No. 26 seed, knocked off third-seeded Agnieszka Radwanska 6-1, 6-2 behind seven breaks. Kuznetsova took nearly two-thirds of the points off Radwanska's serve and had 27 winners to 11 for Radwanska.
Sixth-seeded Samantha Stosur advanced to the fourth round with a 6-3, 6-3 decision over Nadia Petrova and No. 10 Angelique Kerber capped the long day of tennis with a 4-6, 6-3, 6-2 win over 18th-seeded Flavia Pennetta.
No. 15-seeded Dominika Cibulkova stopped Maria Jose Martinez Sanchez 6-2, 6-1 in just more than an hour.
No. 21-seeded Sara Errani defeated No. 13 Ana Ivanovic 1-6, 7-5, 6-3 and Petra Martic eliminated 29th-seeded Anabel Medina Garrigues 6-2, 6-1 in a pair of upsets.
Unseeded Sloane Stephens earned her way into the Round of 16 with a 6-3, 6-2 victory over Mathilde Johansson. The players combined for 61 unforced errors but Stephens had the edge with 26 winners and seven breaks.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys