A runner-up last year, Cornet beat Chanelle Scheepers of South Africa 6-3, 6-1, while Bouchard move into the semi-finals of a tournament for the first time by defeating Anna Tatishvili of Georgia 6-3, 6-2.
Rain interrupted play at 5-1 in the second set of Cornet's match. But the Frenchwoman picked up where she left off, firing a backhand pass to clinch the victory and next play Bouchard.
Bouchard, who entered the top 100 last month, broke Tatishvili twice in each set.
The Strasbourg International is a warm-up tournament for the French Open, which starts Sunday.
Results from the fifth day of the Strasbourg International Women's Singles WTA tournament (x denotes seeding):
Quarter-final:
Alize Cornet (France x3) beat Chanelle Scheepers
(South Africa x7) 6-3 6-1
Eugenie Bouchard (Canada) beat Anna Tatishvili (Georgia) 6-3 6-2