Draper reached his first Masters 1000 quarter-final, one day after defeating Stefanos Tsitsipas in his maiden victory over a top 10 player, when 17th seed Gael Monfils retired hurt in Montreal in the second set with the Brit up 6-2 0-2.
The world number 82, who came through qualifying to reach the main draw, frequently relied on his speed and reach as he claimed a fast first set.
Monfils collapsed not long after securing a break early in the second, prompting Draper to cross the court to help him into a chair as the Frenchman cried out and clutched at his right leg.
Draper, 20, will face either Jannik Sinner or Pablo Carreno Busta next, with the winner of that encounter meeting Evans if the Brit wins his quarter-final.
The Birmingham product continued his run of upsets with a 7-6 (7) 1-6 7-5 dismantling of American 10th seed Taylor Fritz, one day after he knocked out eighth-ranked Andrey Rublev.
Evans roared “yes” and held his arms aloft after qualifying for the tournament’s quarter-finals for the first time, setting up a match against Tommy Paul of the US.
Hometown favourite Felix Auger-Aliassime avenged his weekend semi-final loss to Cameron Norrie in Los Cabos with a 6-3 6-4 effort over the British number one.
The ninth-ranked player took just one hour and 12 minutes to defeat the 26-year-old, much to the delight of the roaring Montreal crowd, and set up a quarter-final against Casper Ruud.