Draper, who won the Indian Wells title last month, slipped to a 6-3, 6-7(6), 6-4 loss as he failed to reach a first Masters quarter-final on clay.
Davidovich Fokina will take on Australia's Alexei Popyrin in Friday's quarter-finals.
The world number 42 is into the last eight in Monaco for the third time, with his best performance a run to the 2022 final which included a shock victory over Novak Djokovic.
Davidovich Fokina edged a hard-fought opening set, which featured three breaks of serve.
Draper saw three break points slip away in the ninth game, with Davidovich Fokina taking his second set point.
The players traded breaks in the second set before Draper saved a match point en route to levelling the contest in a tie-break.
But the Briton, playing in the third round of a Masters tournament on clay for the first time, slipped 2-0 behind early in the decider.
It proved to be a topsy-turvy third set, though, with the pair both broken twice inside the first six games.
Davidovich Fokina delivered the decisive blow in the 10th game to secure his first win over a top-10 opponent on clay since 2023.
Popyrin, the world number 27, later defeated 2024 runner-up Casper Ruud 6-4, 3-6, 7-5.
Musetti, seeded 13, beat Davis Cup team-mate Berrettini 6-3, 6-3 and then talked of "the difficulty of playing a friend, a mate, a teammate."
"This is a special court for me and also for Matteo. We share a lot of moments here in Monte Carlo where we live," Musetti said. "Definitely one of the toughest challenges for me."
World number 10 Alex de Minaur crushed former US Open winner Daniil Medvedev 6-2, 6-2.
"I'm finding my feet on clay," said De Minaur of Australia. "It's taken me a while in my career. I can be dangerous on this surface."
French youngster Arthur Fils eased into his third successive Masters quarter-final this season with an impressive 6-2, 6-3 thrashing of 2023 champion Andrey Rublev.
Fils will next face second seed Carlos Alcaraz, who eased past Daniel Altmaier, 6-3, 6-1, on Thursday.
The 20-year-old Fils is aiming to reach the semis of a Masters event for the first time after losing in the last eight at both Indian Wells and the Miami Open this year.
"If it's him, it's going to be the first time I'm going to play him," Fils said of a potential meeting with Alcaraz.
"It's going to be a good experience. He's a big champion. He won four Grand Slams already. I didn't win a single one."
