Draper defeated Frenchman Gael Monfils, who was forced to retire hurt in the second set, while Evans overcame 10th seed Taylor Fritz.
The British pair are now on course to meet in the semi-finals in Montreal if they can both win their respective last-eight encounters.
After claiming his first victory against a top-10 player in Stefanos Tsitsipas earlier this week, Draper will now also make his debut in a Masters 1000 quarter-final after beating 17th seed Monfils.
“On Tour, sometimes it’s your week,” he said of his breakthrough few days. “If you train hard every day and keep your head down, I suppose these things happen. I’m very happy to be in the quarter-finals here. It means a lot.”
Draper, who is ranked 82nd in the world, took the first set 6-2 but was down 0-2 in the second when Monfils was forced to retire.
Monfils collapsed not long after securing a break and Draper crossed the court to help him to his chair, with the Frenchman clutching his right leg in pain.
“I guess to be in the quarter-finals is nice for myself, but it’s never how you want it to happen,” Draper added.
The 20-year-old Brit will now face Pablo Carreno Busta in the quarter-finals.
Evans, meanwhile, continued his fine form last night by defeating Fritz – who knocked out Andy Murray in the first round – 7-6 (7) 1-6 7-5 just one day after overcoming world no8 Andrey Rublev.
Evans will now take on American Tommy Paul in the quarter-finals after he defeated 13th seed Marin Cilic in straight sets.
It was not a clean sweep of British wins overnight, however, as Norrie was knocked out by Felix Auger-Aliassime.
Backed by plenty of support from his hometown crowd in Montreal, Auger-Aliassime defeated Norrie 6-3 6-4 to avenge his semi-final loss to the British No1 in Los Cabos last week.
Wimbledon finalist Nick Kyrgios will join Auger-Aliassime in the quarter-finals and the pair are on course to meet in the semis.
Kyrgios put in a stunning display to defeat fellow Australian Alex de Minaur in straight sets, 6-2 6-3, after beating world no1 Daniil Medvedev in the previous round.
Big-serving Hubert Hurkacz awaits Kyrgios in the quarter-finals, while Auger-Aliassime will take on Casper Rudd.