Djokovic produced a clinical display to put away Frenchman Jo-Wilfried Tsonga 6-3 7-5 6-4 in a repeat of the 2008 final at Melbourne Park.
It was the 31-year-old Serb's 16th consecutive win on the Grand Slam stage and he will next face Canadian youngster Denis Shapovalov.
Halep was cruising to victory after leading by a set and a break but Kenin roared back to edge the second set in a tiebreak.
Despite going down a break in the decider, Halep kept her composure to prevail 6-3 6-7(5) 6-4.
It is the first time since Wimbledon in 2009 that all of the women's top eight seeds have reached the third round at a Grand Slam.
Serena Williams needed just 70 minutes to swat aside Canada's Eugenie Bouchard 6-2 6-2 as she continued her quest for a record 24th Grand Slam title.
Twice Grand Slam winner Garbine Muguruza overcame Britain's Johanna Konta 6-4 6-7(3) 7-5 in a marathon match that finished after 0300 local time. The Spaniard will next face two-times finalist Venus Williams.
Alexander Zverev survived a scare as he held off Frenchman Jeremy Chardy in a 7-6(5) 6-4 5-7 6-7(6) 6-1 victory.
