The eighth-seeded Musetti's main trouble was intermittent rain on court Simonne Matthieu as he set up a meeting with Argentine Mariano Navone or Reilly Opelka of the U.S.
"It has been a tremendous clay season so far, but I want to go higher. I've never made it past the fourth round here," said Musetti, who reached the Monte Carlo Masters final and the last four in Rome and Madrid to make his first foray into the top 10.
"I'll have to take my chances."
A Galan double fault in the third game handed Musetti the first break, and the Italian opened a 5-2 lead after aggressive play helped him steal his opponent's serve again.
The Colombian pulled a break back in the eighth game as Musetti's serve wobbled. But Musetti saw off another break as he served for the set, which he bagged with a commanding smash.
He then stepped up a gear, allowing Galan only five points in the second set and while the Colombian offered more in the third, there was no comeback as Musetti wrapped it up on serve.