This was a clash between the two highest-ranked players in the field at the event for the best eight men of the season aged 21 and under, with both having lost to in-form Swiss Dominic Stricker and beaten Tseng Chun-hsin to set up a decider.
Italian Musetti, ranked 23 to Draper's 41, had the home crowd on his side but the fast-rising British star was on top from the start in the first-to-four-games format and eased to a 4-1 4-0 4-3 (3) victory.
Musetti needed treatment to his right hand after falling early in the second set and made a fight of things in the third but Draper dominated the tie-break to set up a semi-final meeting with American Brandon Nakashima.
Draper said on atptour.com: "Yesterday I played a bit better than my first match and then today I really hit some form. I played well, served great. Lorenzo is a great player and he may have been a bit tired after his match yesterday but I am glad with the way I played today."
Stricker will take on Czech Jiri Lehecka in the other sem-final.