Sixth-seed Cornet, who won the event in 2016, overcame a fall and grazed knee to win 6-1 6-4 on Thursday and is yet to drop a set at the tournament.
"I fell pretty hard on the hand, but I think it's all good," Cornet said.
The Frenchwoman served for the match at 5-2 in the second set but was broken as Minnen struck some clean winners.
"To finally close it out at 5-4, I was pretty tight, so I'm happy I could finish the match. To be in the semi-final is a great start to the year," Cornet said.
Cornet will meet Sofia Kenin for a place in the final, with the 20-year-old American continuing her impressive run by beating veteran Kirsten Flipkins 7-5 7-5.