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 second Monday is traditionally a feast of top names, with the last 16 in both men's and women's singles being played, but organisers were playing catch-up this year with several third-round matches still to be completed.
Swiss Stanislas Wawrinka, one of the victims of soggy Saturday when his match against Uzbekistan's Denis Istomin was washed out, made up for lost time with a rapid 6-3 6-3 6-4 win completed just before a heavy rain shower arrived.
Japan's Kei Nishikori was detained even less as the 10th seed needed only four games to complete a 3-6 6-3 4-6 7-6(4) 6-4 victory over Simone Bolelli after their match had been suspended because of bad light on Saturday.
Spain's Feliciano Lopez also belatedly reached the last 16, beating big-serving American John Isner in a four-set match that predictably included three tiebreaks.
Isner's defeat ended US interest in the singles after teenager Madison Keys was forced to withdraw on Monday with a leg injury before resuming her third-round match against Kazakhstan's Yaroslava Shvedova.
They had been locked at 6-6 in the second set when darkness fell on Saturday, after Shvedova had won the opener.
Shvedova will play last year's runner-up Sabine Lisicki in the fourth round after the German claimed a stop-go 6-4 3-6 6-1 victory over 11th seed Ana Ivanovic.
GRIT AND POWER
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 quarter-finals if Roland Garros champion Maria Sharapova beats Angelique Kerber later on Monday.
Defending men's champion Andy Murray, who conceded only 19 games on his way to round four, was back in action against giant South African Kevin Anderson before top seed Novak Djokovic's top-of-the-bill clash with charismatic Frenchman Jo-Wilfried Tsonga. (Editing by David Goodman)