Defending champion Federer was at his graceful best in a routine victory over Richard Gasquet, while former world number one Djokovic was hampered by a hip complaint yet nevertheless won in straight sets against Albert Ramos-Vinolas.
Berdych, who has played in the semi-finals twice in Melbourne, will face Italian journeyman Fabio Fognini in their first Grand Slam match in Monday's fourth round.
Kerber, the 2016 champion, is so unassuming and wary of the limelight that she never seemed less happy than when ranked No 1.
It took Fabio Fognini almost eight years, but he's finally avenged a fourth-round loss to Julien Benneteau at Wimbledon.
Their 2010 match at the All-England Club was their only previous meeting.
Halep has been travelling under the radar somewhat at Melbourne Park as she works her way towards a maiden Grand Slam title.
"I knew I wanted to use the medical timeout because I needed it, but at the same time I knew it was nothing major that can potentially raise the question mark on whether I can continue playing or not", he added.
The 97th-ranked Sandgren got past German Maxmilian Marterer 5-7, 6-3, 7-5, 7-6 (7/5) in just under three hours on Show Court Two. There were 11 service breaks and two medical timeouts - for Davis to get treatment on both feet - before Halep converted on her first match point.
Halep will next play Naomi Osaka, who beat 18th-seeded Ash Barty 6-4, 6-2.
Sixth seed Karolina Pliskova also had a tough test, beating fellow Czech Lucie Safarova 7-6 (8/6), 7-5. Garcia will play Madison Keys in the fourth round.
It took Djokovic two hours and 21 minutes to earn his fifth win over Viñolas in as many career matches.
After his gruelling battle in the heat against Nikoloz Basilashvili, conditions are forecast to be much cooler and Edmund will look to set up a clash with either third seed Grigor Dimitrov or Nick Kyrgios, who meet in the big evening match.
In mixed doubles play Canada's Gabriela Dabrowski and Mate Pavic of Croatia are moving on.
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