The day's final match saw current world number two and defending champion Zverev lose his Rome crown in addition to his ranking as he went down to home hope Musetti 7-6(1), 6-4.
After trading breaks earlier in the set, Zverev looked set to seal the opener after breaking Musetti at 5-5 and then leading on his next service game 40-0.
But the Italian improbably saved four set points to force a tiebreak, which he swept 7-1 in front of a raucous centre court.
An angered Zverev argued with the umpire in the second set but kept his cool on the court until 4-4.
The German had looked the more likely to make a breakthrough until the ninth game when a stunning backhand passing shot gave Musetti a break point, which he duly converted before holding to 30 to reach his third semi of the year on clay.
Third seed Alcaraz, earlier, had produced what he called one of his best displays of the season to book a last-four date with the man he beat in Monte Carlo last month.
The Spaniard's impressive win over Jack Draper kept alive his chances of a third title of the season and also moved him up to number two in the world rankings behind Jannik Sinner.