Italian 25th seed Musetti defeated 13th-ranked Taylor Fritz of the United States 3-6, 7-6 (7/5), 6-2, 3-6, 6-1 and will face seven-time champion Novak Djokovic for a place in the final.
"It's a magic day," said the 22-year-old. "I have to say after the birth of my son, I will put this in the second position today.
"For my career, is the best day of my life."
Musetti and his girlfriend Veronica Confalonieri became parents to son Ludovic in March this year.
Now Musetti has another special day on his agenda -- facing Djokovic in Friday's semi-finals at the All England Club.
Djokovic, chasing a record 25th Grand Slam title, was earlier handed a free pass into a 13th Wimbledon semi-final when Alex de Minaur withdrew from their scheduled last-eight tie with a hip injury.
Djokovic has a 5-1 winning record over Musetti, including a come-from-behind five-set victory at the French Open last month.
That third-round match set the record for the latest finish at Roland Garros, with Djokovic completing victory at 3:07 in the morning.
Musetti also took Djokovic the distance in their first meeting at Roland Garros in 2021, when the Italian won the opening two sets of their last-16 tie before injury scuppered his hopes in the decider.
"Djokovic is a legend everywhere. We've played lots of times and I expect a huge fight," said Musetti.
"It's the toughest challenge in tennis but I am an ambitious guy and I like to be challenged. I will give 100 percent."
The Italian added: "After matches with 'Nole', I always learn a lesson.
"That last match against him was stressful. Against him you are more stressed because he's probably the best player ever or one of the best players ever."
On Wednesday, Musetti came back from dropping the first set for the third time at Wimbledon this year after Fritz broke in the fourth game.
The Italian, who plays with a one-handed backhand, levelled on a fourth set point in the second set tie-break and was dominant in the third, breaking the American in the first and fifth games.
Fritz had come back from two sets down to knock out Alexander Zverev in the last 16 but showed no sign of immediately wilting, levelling the contest in the fourth set.
However, Musetti was undaunted, sweeping into a 5-0 lead in the decider on his way to victory.
Fritz also lost a five-set quarter-final at Wimbledon to Rafael Nadal in 2022, with the Spaniard winning the final set tie-break.
"I think the one two years ago hurt even more because I was within points of winning," said the American.
