Frenchwoman Cornet caused a sensation on Saturday by removing five-times champion and pre-tournament favourite Williams, but the fresh-faced Bouchard proved to be made of sterner stuff as she edged to a 7-6(5) 7-5 victory.
"That's cool. I didn't know that," the 20-year-old said of becoming the first Canadian to reach a Wimbledon quarter-final. "Another little historic thing for Canada."
The match was halted after five games to allow the Centre Court roof to slide shut as more rain hit the championships after Saturday's lengthy disruptions.
"It was an honour to play under you @WimbledonRoof" Bouchard, the new golden girl of tennis, tweeted later.
Murray began his fourth-round match against giant South African Kevin Anderson under cloudy skies, building up a commanding lead, but after more rain prompted another roof closure he was given his stiffest examination so far.
The third seed even came within a point of conceding his first set of what has been stress-free tournament for the home favourite before sealing an impressive 6-4 6-3 7-6(6) victory to reach the quarter-finals for a seventh consecutive year.
The 27-year-old is on a semi-final collision course with top seed Novak Djokovic, who eased past Frenchman Jo-Wilfried Tsonga 6-3 6-4 7-6(5), but must first negotiate his way past in-form Bulgarian Grigor Dimitrov, who beat Argentine Leonardo Mayer.
"I knew I was going to get tested at some stage," Murray, who had arrived in the second week having dropped a mere 19 games, told reporters. "Today I was pushed. Later on in the third there were some tight moments."
Dimitrov will pose a threat to Murray's 17-match winning streak at Wimbledon (including the 2012 Olympics), especially as he has already beaten the Scot this year.
"My job isn't over yet," the 23-year-old warned after continuing his best Wimbledon run.
Djokovic's match against 14th seed Tsonga, a repeat of their 2011 semi-final, had been given top-billing, but the Serb was at his ruthless best in a scintillating display under the roof to set up a last-eight meeting with Croatia's Marin Cilic.
The second Monday at the All England Club is traditionally last 16 day in both men's and women's singles, but several third-round matches had been held up by rain at the weekend.
Swiss Stanislas Wawrinka, one of the victims of soggy Saturday when his match against Uzbekistan's Denis Istomin was postponed, made up for lost time with a rapid 6-3 6-3 6-4 win completed just before a heavy rain shower arrived.
The Australian Open champion wasted little excess energy against the bespectacled Istomin, but he grumbled about the schedule that leaves him facing three matches in as many days.
"They just say what's going to be the schedule and that's it," Wawrinka said of the decision not to play his third-round match when the rain eventually stopped on Saturday.
"They're not going to change anything. They don't listen to the player. They just do what they think is good for them."
Djokovic had some sympathy.
"I understand why Wawrinka was complaining," he said. "We have this tradition here of no play on middle Sunday.
"So I think we have to rethink about this Sunday in between because the day was beautiful. There was no rain and the whole day there was no match played."
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 and 86 aces.
Isner's defeat left the United States without a representative in the last 16 of the men's or women's singles for the first time since 1911 after teenager Madison Keys withdrew injured before resumption of her held-over match against Kazakhstan's Yaroslava Shvedova.
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 against 11th seed Ana Ivanovic.
Japan's Kei Nishikori, the 10th seed, completed a 3-6 6-3 4-6 7-6(4) 6-4 third-round victory over Simone Bolelli after their match had been suspended at the weekend.
GRIT AND POWER
The defeat of Williams blew a gaping hole in the women's draw and Canada's Bouchard exploited it with a performance of style, grit and power against 25th seed Cornet.
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 to go even further on the slick Wimbledon lawns.
She looked like she would be stretched into a third set when she trailed Cornet 5-3 in the second, but she continued her fearless shot-making with abandon 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 finally succumbed when she fired a backhand long.
"I've proved to myself I can play on the big stage as well. I've played on centre courts of most of the slams; big moments, big matches. I'm proud of the way I can handle it," the supremely confident Bouchard told reporters.
She could now get another crack at Maria Sharapova, who beat her in the French Open semis this month, if the Roland Garros champion beats Germany's Angelique Kerber in a fourth-round match that fell foul of the rain.
The bottom half of the women's draw is complete, however, and is dominated by Czechs.
Former champion Petra Kvitova moved stealthily on with an easy win against China's Peng Shuai and will play compatriot Barbora Zahlavova Strycova, who frustrated former world No.1 Caroline Wozniacki's hopes of a first Wimbledon quarter-final.
Polish fourth seed Agnieszka Radwanska was thrashed 6-3 6-0 by Russian Ekaterina Makarova who will face another Czech, 23rd seed Lucie Safarova who dismantled Tereza Smitkova.