The Spaniard was beaten to the punch in the opening set but raised his game to claim a comfortable 3-6, 6-3, 6-3, 6-3 win to the delight of the Centre Court crowd.
He will now face a battle of the 20-year-old's against Denmark's sixth seed Holger Rune for a place in the semi-finals.
Berrettini came out firing with his serve and forehand rocking Alcaraz but after an injury-plagued year he could not hope to maintain that intensity.
Alcaraz was relentless from then on, running former runner-up Berrettini ragged at times.
He broke serve in the third game of the second set on the way to levelling the match and did the same in the third set as he picked up speed.
A delay to close the roof midway through the fourth set as the light faded offered Berrettini some respite but Alcaraz quickly completed the win after the resumption.
"I'm hungry for more," Alcaraz said on Centre Court. "It's something I really wanted. I came in this year with that goal - get into the quarter-final, I'm wanting more.
"I want to play the final here and maybe win it."
