Musetti, who was beaten by Novak Djokovic in the semi-finals, prevailed 6-4 1-6 6-3 on Saturday at Roland-Garros.
It marked a historic success for Italy, bringing up the nation's first tennis player to win a medal at the Olympics since Uberto de Morpugo in 1924, with those Games having also taken place in Paris.
While Auger-Aliassime leaves empty-handed from the singles, he did win bronze in the mixed doubles.
Prior to Musetti's triumph, Tommy Paul and Taylor Fritz won in straight sets to claim the bronze medal in the men's doubles, beating Tomas Machac and Adam Pavlasek.
After falling to defeat to eventual gold medallists Matthew Ebden and John Peers in the semi-finals, the American duo dusted themselves down to claim a 6-3 6-4 victory.
Paul and Fritz controlled the opening set, earning a double break in the sixth game and, despite losing the next two, held on to take the early advantage.
The Americans squandered three match points to seal the win, but confirmed their place on the podium in the 10th game, with Team USA taking home two medals from the doubles after Austin Krajicek and Rajeev Ram's silver earlier in the day.
It was not all doom and gloom for Czechia, though, with Machac and Katerina Siniakova having claimed gold in the mixed doubles on Friday.
Alexander Zverev had an altogether more comfortable progression at Ariake Tennis Park, while home favourite Kei Nishikori earned an impressive upset win.
Felix Auger-Aliassime was unable to make the most of two-time defending champion Andy Murray's withdrawal, but it was a good day for Hubert Hurkacz.
Here's the pick of the action from day two of the men's singles.
TSITSIPAS OUT TO EMULATE GRANDFATHER
Greek ace Tsitsipas, a French Open finalist this year, had to dig deep for a three-set win over Philipp Kohlschreiber.
Should he manage to win gold in the Japanese capital he would be matching the feat of his grandfather, who won the football competition representing the Soviet Union in 1956.
"I've never had the opportunity to meet him. But my mom told me stories of his career and how he got it," he said. "He kind of inspires me in a way. I know what kind of athlete he was, with all the achievements and all the trophies. I'm proud of him.
"It's something good, a legacy that is being carried on in the family. I'm happy to be the next in the family to be competing at the Olympics."
ZVEREV LOVING OLYMPICS EXPERIENCE
Fourth seed Zverev coasted past Lu Yen-hsun 6-1 6-3 and spoke of how much he is enjoying being around other German athletes.
"Normally you don't have those guys around that much, you have your friends, of course you have people that are around you, but you don't sleep in the same room as them," he said.
"Yes it is very different but in a way very enjoyable. The Olympics are once every four years, and it’s five years now, so I think everybody is enjoying it and everyone is having the best time that they can."
Nishikori is playing at a fourth Olympics and upset fifth seed Andrey Rublev 6-3 6-4. For the 31-year-old the motivation is simple.
"It's [playing in Japan] something I always dreamed of when I was little," he said.
"Especially now, with the Covid situation, if I can win as many as I can, I think it will bring better news, that's something I'm trying to do this week."
AUGER-ALIASSIME FAILS TO MAKE MOST OF MURRAY ABSENCE
Auger-Aliassime was scheduled to face Murray before the Team GB star pulled out with a quad issue and will instead focus on doubles.
The Canadian was felled by Murray's replacement Max Purcell, though, the Australian winning 6-4 7-6 (7-2).
Fellow seed Gael Monfils also crashed out but Hurkacz was a 6-2 6-4 victor over Luke Saville, while Diego Schwartzman also made it through.
