Under new coach and old rival Andy Murray, the 37-year-old Djokovic needed four sets in both of his opening two matches.
But not this time, swatting aside Czech 26th seed Tomas Machac 6-1, 6-4, 6-4 and letting out a roar at the end.
Djokovic also pointed to his ear and then someone in the Rod Laver Arena crowd.
“I am very hot-headed right now,” said the fiery Serb, but he added: “This is definitely the best match I’ve played in the tournament.”
Djokovic, who needed a medical break in the second set and used an inhaler, faces 24th seed Jiri Lehecka next.
Spain’s Alcaraz, who raced into the last 32 for the loss of just 12 games, suffered a wobble in the third set against Portugal’s unseeded Nuno Borges.
But the third seed, who has won four majors but never been beyond the quarter-finals in Melbourne, regained his focus to ease through 6-2, 6-4, 6-7 (3/7), 6-2.
The 21-year-old knows how he plans to celebrate if he wins the tournament for the first time.
Alcaraz, who is sporting a new buzz cut in Melbourne, vowed to get a kangaroo tattoo if he goes all the way.
“It’s a kangaroo, that’s for sure. That’s my idea,” he said.
Germany’s second seed Zverev also booked his place in the second week with a methodical 6-3, 6-4, 6-4 win over Britain’s Jacob Fearnley.
Zverev has not dropped a set in three matches so far and faces French 14th seed Ugo Humbert.
Jakub Mensik, the Czech teenager who shocked sixth seed Casper Ruud in round two, crashed out in five sets to Spain’s Alejandro Davidovich Fokina.
Men’s defending champion and world number one Jannik Sinner plays Marcos Giron of the United States in the third round on Saturday.
‘It sucks’
Top-ranked Sabalenka was made to work hard behind a faltering serve before beating Denmark’s Clara Tauson in the third round on Rod Laver Arena.
Sabalenka was broken four straight times at the start of the match but found her groove to win 7-6 (7/5), 6-4 after more than two hours of attritional tennis.
“I’m just super happy that I was able to just stay in the game and I was able to push myself, honestly, to the limit to get this win,” said Sabalenka, who is chasing a third Melbourne title in a row.
Her win puts her into a clash against highly rated 17-year-old Andreeva, the 14th seed, who beat Poland’s 23rd seed Magdalena Frech in three sets.
Andreeva beat Sabalenka in the quarter-finals at Roland Garros last year.
Two-time Melbourne champion Osaka was on course to clash with third seed Gauff in the next round.
But the former world number one needed treatment on her stomach during the first set against Belinda Bencic and called it quits after losing the opener on a tiebreak.
Osaka, who retired from the final in Auckland with an abdominal injury in the lead-up to Melbourne, said: “If I could have served, then I could have potentially won and I could have maybe gone far in the tournament.”
She added: “It sucks”.
Bencic will face Gauff after the American eased to a 6-4, 6-2 victory over Canada’s 30th seed Leylah Fernandez.
Gauff is yet to drop a set this season in eight matches, having won all five of her singles at the United Cup.
Seventh seed Jessica Pegula, the United States Open finalist last year, was a surprise straight-sets loser to Serbia’s Olga Danilovic.
Second seed Iga Swiatek plays Emma Raducanu, a clash between former US Open champions, in the pick of the women’s matches on Saturday.
