World number one Djokovic made a poor start and endured a second-set wobble before coming through 7-5, 6-3 against the 24th-ranked Musetti, the man who beat him at the same stage 12 months ago.
"I don't think I am still at my top level but it was a great test today against a great player, a very talented player," Djokovic said.
"I am really glad to overcome the challenge and look forward to the next one."
Djokovic goes on to play Alex de Minaur, the 11th seed from Australia, in the last eight.
The Serbian top seed dropped serve in the opening game against Musetti before working his way back to level at 4-4, and then snatched the set on a double fault by his opponent.
Djokovic broke twice early in the second set either side of losing his own serve to surge 4-1 ahead. Musetti clawed back to within a game but Djokovic broke again for 5-3 before closing out the win.
Djokovic has struggled to find his best form in recent times in Monte Carlo. He has not won the tournament since the last of his two titles in 2015, failing to advance beyond the quarter-finals in seven subsequent appearances.
Australian Open champion Sinner improved his record to 24-1 this season with a 6-4, 6-2 win over Jan-Lennard Struff.
The world number two broke at 4-4 en route to pocketing the first set and won nine of the final 11 games to sweep to victory.
Sinner awaits the winner of the match between Holger Rune and Grigor Dimitrov, who only completed their second-round matches earlier in the day after rain brought a premature end to play Wednesday.
Rune beat Sinner in the semi-finals last year before finishing runner-up to Andrey Rublev.
Medvedev tore into the officiating for the second day running as he was beaten 6-3, 7-5 by Karen Khachanov.
Medvedev was hit with a point penalty to start the final game after berating chair umpire Carlos Bernardes and then became embroiled in a heated discussion with the tournament supervisor.
The world number four had already been warned after hurling his racquet when he double-faulted to lose his service game and leave Khachanov a game away from victory.
Medvedev, who took issue with two line calls in his second-round win on Wednesday, felt aggrieved a Khachanov forehand was not called out during a rally that resulted in the latter earning two break points at 5-5.
The arrival of the physio to treat a finger bleed at the ensuing changeover tipped Medvedev over the edge, with the Russian yelling "did I ask for the physio!" at the umpire.
'Open your eyes'
He then demanded of the supervisor "who will take responsibility?" for the decisions.
Television replays appeared to show the shot in question from Khachanov had landed wide of the court.
"It's second day in a row. Guys, open your eyes. Do something. It's out," raged Medvedev.
"The mark is out. They don't know how to referee anymore. Who will take action?
"Yesterday the ball is out, it's called in. Who will take action. This ball is out there. Who will take responsibility? It's not my responsibility to referee the matches.
"It's this guy in the glasses (the line judge). He doesn't need glasses because he doesn't see anything. He should not be a referee."
Khachanov's reward is a quarter-final Friday against Stefanos Tsitsipas, who saw off fifth seed Alexander Zverev 7-5, 7-6 (7/3).
De Minaur moved into the last eight with a 6-3, 6-4 victory against fellow Australian Alexei Popyrin.
