The Italian came through a five-set battle with Tomas Machac in the last round but it did not appear to have taken much out of him as he broke Fritz's big serve twice in the opening set.
Musetti played with authority and artistry towards the end of the second set, which he secured by winning three straight games, before breaking early in the third set to heap the pressure on the American.
"Honestly, I feel very proud. I know Taylor very well, we've played many matches, many battles. And last time in Turin he was the winner," Musetti said about the pair's meeting at the ATP Finals last year.
"I came here with a different mentality and I think I made one of my best performances ... I'm very happy. I improved a lot on serve and especially being a little bit more aggressive with the forehand and trying to use my variations.
"My coach always keep telling me to be a little bit more aggressive and try to go for it, to try to take the lead in the rallies. And that's what I did today."
Ninth seed Fritz, who was moved all over the court by his 23-year-old opponent, dug deep to stay in touch but Musetti served out the victory in style for a tantalising meeting with 24-time major champion Djokovic.
The Serb had already advanced to the last eight after his fourth-round opponent Jakub Mensik withdrew due to an abdominal injury on Sunday.
"Novak, we've played many, many times and every time it's a lesson, first of all," said Musetti, who has won only one of his 10 meetings with the 38-year-old.
"It's such an honour to share the court with him, and every time I leave the court with something I really think is helping me to try to win against him.
"He didn't have to play today, so I'm pretty sure he won't be tired. Hopefully the rhythm I have right now with the great match today will bring me luck for the next one.
"I feel ready to try to push him to the maximum."