Minella, playing in her first final on tour at age 32, was seeking to become the third mother to win a WTA singles title in the past year.
The 48th-ranked Cornet did not drop a set all week in the Swiss Alps clay-court event to win her first title since January 2016, at the Hobart International in Australia.
Cornet won five of the 20 break points she held against the 226th-ranked Minella.
The 28-year-old Frenchwoman clinched the first set with a drop-shot winner, and saved two set points in the second-set tiebreaker.
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