Draper battled to a 6-3, 7-6 victory over Monte Carlo runner-up Musetti, hanging in there during a second set that featured some mesmerising tennis from the Italian before snatching the win in the tiebreak.
The 23-year-old lefty fired 20 winners against 19 unforced errors in what was an extremely high-quality two-hour showdown with the Italian.
As a result, the Brit improved to 4-0 in career meetings with Musetti and advanced to his second Masters 1000 final of the year.
"I felt like both of us, our quality didn't drop from the first ball. Credit to Lorenzo, he's honestly playing so good on the clay," said Draper, the champion in Indian Wells in March, who's set to enter the top five next week.
"I've played him on the hard and on the grass in the juniors, growing up with him. But on the clay he's a different beast and so, to get this win, on this court, in this stage, in the semi-finals of this competition, honestly it means so much to me."
In the final, Draper will face Casper Ruud, who beat Francisco Cerundolo earlier in the day.
"Casper is such an experienced player, someone who has made finals of two Grand Slams, is just a complete pro, and someone who always gives his best and is so tough to beat," said Draper.
"He's in the final, he's playing well, and I'm going to have to bring my A-game for sure."