The 78th-ranked Goffin entered the event on a wild card. Since losing to Andy Murray in the first round at Wimbledon, Goffin has been enjoying an unbeaten streak of 19 matches, including three challenger events he won.
Goffin dropped serve twice in the opening set but dominated play afterward, converting his fifth match point with a return winner on the 104th-ranked Argentine's serve.
Earlier, Thiem beat defending champion Marcel Granollers of Spain 6-4, 7-5 to reach his maiden semifinal.
The 20-year-old Thiem, the youngest player in the top 50, was 3-1 down in the opening set before winning 10 of the next 15 games for a 5-2 lead in the second. He failed to serve out the match at 5-3 but broke the second-seeded Granollers again at 6-5.
Monaco defeated the fourth-seeded Andreas Seppi of Italy 6-2, 3-6, 6-3. Both quarterfinals were moved from Thursday because of rain.
The Associated Press contributed to this report.