The Italian had won the opening two sets 6-4 6-3, prompting 10-time Open champion Djokovic to call a medical timeout to get treated for blisters on his right foot.
The revitalised Djokovic then broke Musetti to go ahead 2-1 in the third set, only for the Italian to call for treatment from the physio on his groin.
A game later, and trailing 3-1 in the third set, the No.5 seed signalled he couldn't continue.
Djokovic conceded that he got lucky, saying that his opponent was headed for victory.
"I don't know what to say, except I feel really sorry for him. He was a far better player. I was on my way home tonight," the 38-year-old said on court.
"These things happen in sport. It happened to me a few times. But being in the quarters of a grand slam, two sets to love up and being in full control. I mean, so unfortunate.Â
"I really wish him a speedy recovery and he should have been a winner today."
Musetti was also forced to retire from the French Open semi-finals last due to injury, having taken the first set from eventual champion Carlos Alcaraz.
Chasing a record-extending 25th major title, Djokovic also benefited from his opponent's misfortune in the previous round when 15th-seeded Czech Jakub Mensik pulled out before the match through injury.
Djokovic advanced to a semi-final against either two-time defending champ Jannik Sinner or American No.8 seed Ben Shelton.
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