Garcia, 75th in the world, qualified first on Thursday by beating Germany's Sabine Lisicki, 804th in the world, 6-3, 7-6 (9/7), before Cornet (44th) took out another German, Angelique Kerber (18th), former world N.1 and defending champion, beaten 6-4, 2-6, 6-1.
In Germany, Garcia has already equaled her best performance this season, during which she reached a semi-final in Lyon last February.
Unhappy Wimbledon finalist in 2013 against the French Marion Bartoli, Lisicki, 32, had received an invitation from the organizers to play in the German tournament.
In the other two quarter-finals, Russian Daria Katsakina, seeded N.1, was surprised by Canadian Bianca Andreescu, crowned at the US Open in 2019, who won 6-4, 6-1 , while Romanian Simona Halep, victorious at Roland-Garros in 2018 and at Wimbledon in 2019, very easily dominated American Amanda Anisimova, yet a great hope of the WTA circuit, 6-2, 6-1.