Berrettini, 23, became just the second Italian man to progress to the US Open quarter-finals on Monday, 42 years after Corrado Barazzutti's run to the last four in New York.
Berrettini, the 24th seed, eased to a 6-1, 6-4, 7-6 (8/6) win over Russia's 43rd-ranked Andrey Rublev to reach the last eight at a Grand Slam for the first time.
"It's something really crazy. I cannot believe it right now. I need a few hours to understand what happened," said Berrettini, who goes on to play French 13th seed Gael Monfils.
"Corrado Barazzutti is the last one that did semi-final here, first one, last one. He's my Davis Cup captain. We're talking every day. He's helped me a lot," Berrettini said.
Barazzutti's run to the semi-finals of the 1977 US Open, a tournament played on clay, came a year after Adriano Panatta knocked out Bjorn Borg on his way to capturing the French Open title.
Panatta was the only player to defeat Borg at Roland Garros, doing so on two occasions.
"I'm coming from Rome, so Adriano Panatta is coming from there. He usually text me a lot," said a smiling Berrettini.
"I have to say he's the first one that told me you're going to serve like 220 kilometers per hour. I was 16 when I heard that. I was, like, 'I don't know. If you say that, I'm going to trust you.'
"For me they're guys that I have to look to. Tennis for sure is different right now. But the mental part of tennis is the same. I'm looking those kind of matches, those kind of mental toughness. Yeah, they're really inspiring me."
Berrettini has gone one step further than at Wimbledon in July, where he was beaten by Roger Federer in the last 16.
"I'm just happy. Seeing my team and family here and crying, for me it's unbelievable."
The Italian overcame a momentary lapse against Rublev as he dropped serve at 6-5 in the third set but sealed victory after two hours and 11 minutes in a tense tie-break.
"I think I did an unbelievable match. I thought Andrey was playing really good and playing with confidence," Berrettini said.
"I was mixing my tennis, using the slice, forehand and serve. I think all my weapons were really good today."