The 25th-seeded Italian's bid to join compatriot and world number one Jannik Sinner as a major champion this year faces a massive hurdle, however, as he faces seven-times All England Club champion Novak Djokovic in the last four.
"I'm really happy about what I've achieved today. Happy about the way I fought until the end," Musetti told reporters.
"I found a way to develop my game set after set. It was really a big reaction from me in the fifth. I probably played my best tennis of the week."
Musetti has beaten Djokovic once in six clashes and dragged the Serb to five sets in their most recent one at Roland Garros.
The 22-year-old showed similar battling qualities against Fritz after going behind early in the match.
After holding in a tight first game, Fritz seized the early break for 3-1 when Musetti netted a backhand, and the American consolidated the advantage in the next game with a powerful forehand winner to set the tone for the first set.
Eastbourne champion Fritz continued to crank up his forehand and serve in glorious sunshine, occasionally drawing gasps from Court One fans with rockets from the baseline, and grabbed the lead in the match with an ace.
After the pair exchanged breaks at the start of the second set, Queen's Club runner-up Musetti pounced again to go 5-3 up, and although Fritz erased the deficit the Italian raised his level to edge the tiebreak.
Musetti mixed up his game with chips and slices and forced Fritz into uncharacteristic errors, easing through the third set on the back of two breaks.
"That's probably something that I say (is) worse for the other guys, not having each shot the same," Musetti said.
"Especially against a good baseliner like Taylor, if you play flat every time, you can't win a point. I mean, I'm playing his game. So probably it was the strategy to try to mix every ball and try to lead the game.
"At certain points of the match I felt from the baseline I could win every point. He wasn't in a good position with all my variations."
DJOKOVIC AWAITS
Musetti, however, handed the American a way back into the match after dropping the next set before lifting his level.
As shadows covered the manicured turf and blustery weather made life tough for the players, Musetti built a 5-0 lead in the decider and finished it off on serve, when Fritz hit a shot long after falling awkwardly at the net in the previous point.
Musetti's reward is a meeting with 24-times Grand Slam champion Djokovic, who went through to a record-equalling 13th semi-final at Wimbledon after Australian ninth seed Alex De Minaur pulled out of their match with a hip injury.
"I've played him many times in different stages, even big ones. I beat him one time in Monte Carlo. I lost the others," Musetti said, before reflecting on their clash at this year's Roland Garros which ended shortly after 3 a.m in Paris.
"With Nole, after the match I always finished with a lesson. The last match was really an intense one from both players and a really stressful match.
"Against him you're probably more stressed because he's probably the best player ever or one of the best ever. You walk on court with a different mentality. If I play a certain way, I could have my shot in the next round."