Berrettini, a finalist in 2021, won the first set but the top seed roared back to win the Centre Court match-up 3-6, 6-3, 6-3, 6-3.
Top-ranked Alcaraz is seen as one of the few genuine threats to defending champion Novak Djokovic, who has not lost a match on Centre Court for 10 years.
But he was given a wake-up call by the 38th-ranked Italian at the showpiece arena, losing the first set after being broken in the eighth game.
The Spaniard, 20, forged a break of his own in the fourth game of the second set, which enabled him to get back on level terms.
A double break in the third set put him in control and he broke again in the eighth game of the fourth set.
He sealed victory when Berrettini, forced to miss last year's Wimbledon after testing positive for Covid, went wide with a backhand.
Alcaraz has little experience on grass in his short career but won last month's tournament at Queen's to set himself up for a tilt at the Wimbledon title.
"I really wanted to play the quarter-final here, coming this year with that goal," said the US Open champion, who reached the fourth round at the All England Club last year.
"It's my dream to play a final here, to win this title one day, so I hope to reach that dream this year but right now it's great to be in the quarter-finals."
