The pair loomed as two of the biggest winners when Novak Djokovic's dramatic deportation blew their quarter of the draw wide open.
Now a deep run into the second week is the Italian's for the taking after he resisted a spirited fightback to secure a 6-2 7-6 (7-3) 4-6 2-6 7-6 (10-5) victory on Friday.
Berrettini appeared headed for a deserved early shower when he took control and went two sets up against 18-year-old Alcaraz; the No.31 seed and youngest player in this year's men's tournament.
But the Wimbledon runner-up dropped the next two sets and lost momentum before suffering a further setback when he rolled his ankle when leading 1-0 on serve in the fifth set.
He called for the trainer and required treatment as Alcaraz drew applause from the crowd for walking over to check on his opponent.
Berrettini recovered and blew one match point on Alcaraz's serve, but benefited from the Spaniard's crucial unforced errors on big points in the deciding tiebreak.
He advanced in an epic encounter that lasted more than four hours.
"I was feeling confident in the third and I felt the momentum was on my side and then tennis ... I got broken," Berrettini said.
"In the fourth (set) my energy wasn't the right one, but it's part of the job.
"When I started the fifth set I started to think just about fighting every point and I feel like every match something has to happen - this is my body - but I fought through that as well."
The win was a slice of revenge for Berrettini, who lost his only previous match against Alcaraz in a third-set tiebreak in Vienna last October.
Berrettini will next meet another Spaniard in No.19 seed Pablo Carreno Busta, who beat American Sebastian Korda 6-4 7-5 6-7 (6-8) 6-3.
It is the first time 25-year-old Berrettini will feature on court in the fourth round at Melbourne Park, after an abdominal strain saw him withdraw before his meeting with Stefanos Tsitsipas of Greece last year.
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