Draper beat Frenchman Gael Monfils, who was forced out injured in the second set, while Evans defeated 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 last eight respective encounters.
After picking up his first victory against a top-10 player in Stefanos Tsitsipas earlier this week, Draper will now also make his Masters 1000 quarter-final debut after beating 17th seed Monfils.
“On tour is sometimes your week,” he said of breaking through in a matter of days. “If you train hard every day and keep your head down, I guess those 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, won the first set 6-2 but lost 0-2 in the second when Monfils was forced to retire.
Monfils collapsed shortly after earning a break and Draper crossed the pitch to help him sit down, the Frenchman clutching his right leg in pain.
“I guess being in the quarter-finals is fine for me, but that’s never how you want it to be,” Draper added.
The 20-year-old Briton will now face Pablo Carreno Busta in the quarter-finals.
Evans, meanwhile, continued his fine form last night beating Fritz – who knocked out Andy Murray in the first round – 7-6(7) 1-6 7-5 just a day after beating world number 8 Andrey Rublev .
Evans will now face American Tommy Paul in the quarter-finals after beating 13th seed Marin Cilic in straight sets.
It was not a clean sweep of overnight British wins, however, as Norrie was knocked out by Felix Auger-Aliassime.
Buoyed by plenty of hometown crowd support in Montreal, Auger-Aliassime beat Norrie 6-3 6-4 to avenge his semi-final loss to the British No.1 in Los Cabos last week.
Wimbledon runner-up Nick Kyrgios will join Auger-Aliassime in the quarter-finals and the duo are set to meet in the semi-finals.
Kyrgios put on a stunning display to defeat fellow Australian Alex de Minaur in straight sets, 6-2 6-3, after defeating world number one Daniil Medvedev in the previous round.
Big servant Hubert Hurkacz awaits Kyrgios in the quarter-finals, while Auger-Aliassime takes on Casper Rudd.