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NIneteen-year-old Musetti trailed 6-3 in the third-set tie-break after letting an earlier match point slip through his grasp.
But Evans could not convert any of his first three match points and, after Musetti let another one go begging, the Briton failed to take advantage of his fourth opportunity.
Evans' profligacy eventually proved costly, Musetti converting his third match point with a well-executed forehand drop shot to clinch a 6-1 1-6 7-6 (10-8) triumph.
Seven of Musetti's 12 ATP Tour level wins have now come against players in the top 35 of the rankings.
"The secret for me is not to think about [their ranking], to just play them like other opponents," said Musetti, quoted on the ATP website. "I try to play my game… and try to play as good as I can.
"That is what I did. With them, I always play good. I don't know why. I have no pressure and I feel free. I have no pressure to win and today was the same."
Sixth seed John Millman lost in straight sets to Laslo Djere, but third seed Lorenzo Sonego eased past Gilles Simon 6-4 6-1 to set up a meeting with Yannick Hanfmann, who overcame Marco Cecchinato.
At the Andalucia Open, the seeds tumbled out in Thursday's action.
Second seed Fabio Fognini was thrashed 6-2 6-1 by Jaume Munar, Alejandro Davidovich Fokina lost in straight sets to Ilya Ivashka and sixth seed Feliciano Lopez was undone by a comeback from Carlos Alcaraz.
Seventeen-year-old Alcaraz recovered after losing the first set to beat veteran Lopez 4-6 6-2 6-4. Bucking the trend of the day, third seed Casper Ruud prevailed 6-3 6-2 against Gianluca Mager.