21 year old Draper continued to brush off a pre-tournament injury with a 6-4 6-2 3-6 6-3 win over American Michael Mmoh.
Draper who was born in Sutton and raised in Ashtead is a former Wimbledon junior finalist and has been as high as number 38 in the World.
He now he faces number 8 seed Audrey Rublev.