Azarenka ruthlessly disposed of Austria's Yvonne Meusberger 6-1 6-0 in just 60 minutes in the night session while earlier Sharapova overcame a nervous finish to beat Alize Cornet 6-1 7-6 (8-6) on a cooler Melbourne day.
Azarenka was simply awesome against Mesuberger as the grunting Belarusian, who makes nails scratching a blackboard sound like Mozart's Symphony No.31, used her dominant power game to give the world number 49 little chance to express herself.
The two-time defending champion raced to a 5-1 lead in the first set and, after wasting two set points with unforced errors, would make no mistake at her third opportunity as Meusburger sent a forehand long to lose the first set in 37 minutes.
And it was more of the same in the second set as the world number two broke the helpless Meusberger a further three times on the way to 'bagelling' her opponent and keeping her bid to become the first woman in 15 years to win three consecutive Australian Opens on track.
Making the win by Azarenka even more impressive was that it sounded like she was choking on a bee for the entirety of the match and attempted to cough it up whenever she hit a ball.
Azarenka will now surely face a much sterner test in the form of rising American star Sloane Stephens in the round of 16 after quadrupling Meusberger for winners (20-5) and breaking her serve seven times as well as winning 71 percent of first-serve points and 73 percent of return points against the Austrian's first serve.
Sharapova, who was pushed by Italian Karin Knapp in searing heat during a three-and-a-half-hour second-round match, had more troubles on her serve.
The third seed dictated the contest against Cornet from the back of the court for the most part, winning in one hour and 51 minutes.
Sharapova is now into the fourth round at Melbourne Park for the fourth straight time, but will need to improve on serve after producing eight more double-faults to take her tournament-high tally to 29.
The four-time grand slam champion will face Slovakian 20th seed Dominika Cibulkova, who thrashed Spain's Carla Suarez Navarro 6-1 6-0.
A slow start by Cornet helped Sharapova to a 4-0 lead and the first set was over in 32 minutes when the Frenchwoman committed her ninth unforced error.
Cornet won just seven points on the Sharapova serve during the set, with the 2008 champion pushing her well behind the baseline.
But Cornet fought hard to start the second, recovering from 0-40 in the third game to hold for 2-1.
Cornet broke in the next game with a wonderful drop volley, but was broken to love - completed with a double fault.
Sharapova lifted a gear and Cornet had no answers, until the Russian struggled to close after taking a 5-3 lead.
Cornet saved a match point in the ninth game and levelled at 5-5 thanks to Sharapova's eighth double fault.
The pair traded breaks again, Sharapova saved a set point in the tie-break before Cornet sent a forehand long to end a 76-minute set.
Cibulkova only needed 59 minutes to brush past 16th seed Suarez Navarro, who committed 23 unforced errors.
Romanian 11th seed Simona Halep reached the last 16 in Melbourne for the first time after a 6-1 6-4 win over Kazakh Zarina Diyas.
Awaiting Halep in the fourth round is Serbian Jelena Jankovic after the eighth seed defeated Japan's Kurumi Nara 6-4 7-5 in an error-riddled encounter.
Last year's semi-finalist Stephens overcame a slow start to claim a 7-5 6-4 win over Ukrainian teenager Elina Svitolina.
The American 13th seed trailed 5-3 in the first set but recovered after 19-year-old Svitolina failed to serve it out.
Stephens hit 24 winners, including 14 off the forehand wing, to progress in one hour and 18 minutes.
Meanwhile, Polish fifth seed Agnieszka Radwanska survived a scare, battling past Anastasia Pavlyuchenkova 5-7 6-2 6-2.
Next up for Radwanska is big-hitting Spaniard Garbine Muguruza, who continued her good form with a 4-6 7-5 6-3 win over Danish 10th seed Caroline Wozniacki.
The Venezuela-born Hobart International champion overpowered Wozniacki and while her unforced error tally (56) was high, 33 winners did the damage.
3-Maria Sharapova (Russia) beat Alize Cornet (France) 6-1 7-6 (8-6)
8-Jelena Jankovic (Serbia) beat Kurumi Nara (Japan) 6-4 7-5
11-Simona Halep (Romania) beat Zarina Diyas (Kazakhstan) 6-1 6-4
20-Dominika Cibulkova (Slovakia) beat 16-Carla Suarez Navarro (Spain) 6-1 6-0
13-Sloane Stephens (U.S.) beat Elina Svitolina (Ukraine) 7-5 6-4
5-Agnieszka Radwanska (Poland) beat Anastasia Pavlyuchenkova (Russia) 5-7 6-2 6-2
Garbine Muguruza (Spain) beat 10-Caroline Wozniacki (Denmark) 4-6 7-5 6-3
2-Victoria Azarenka (Belarus) beat Yvonne Meusberger (Austria) 6-1 6-0