Frenchwoman Cornet caused a sensation on Saturday when she sent five-times champion and pre-tournament favourite Williams packing but Bouchard proved to be made of sterner stuff as she edged to a 7-6(5) 7-5 victory.
The match was interrupted after five games to allow the Centre Court roof to close when more rain hit the championships after Saturday's serious disruption.
The defeat of Williams blew a large hole in the women's draw and Bouchard exploited it with a performance of grit and power against 25th seed Cornet to become the first Canadian to reach a Wimbledon singles quarter-final.
Having reached the semi-finals at both the Australian Open and French Open this year, Bouchard arrived at Wimbledon with the likes of former men's champion John McEnroe tipping her as his dark horse for the women's title.
She looked like she would be stretched into a third set when she trailed 5-3 in the second, but she continued her attacking style to claw back the deficit.
Serving at 5-6, Cornet tried in vain to hold off the Canadian trailblazer, rescuing one point after a nasty tumble, but fired a backhand long to succumb.
"I am really excited and proud of the way I performed," Bouchard told the BBC. "It wasn't easy, Alize is a great player and got so many balls back, so I had to work really hard.
"I am very excited about reaching the quarter-finals, but I need to stay focused and not get distracted."
Bouchard could face a repeat of her French Open semi-final in the last 16 if Roland Garros champion Maria Sharapova beats Angelique Kerber later on Monday.
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