Draper claimed the first set inside 32 minutes before Cerundolo provided a sterner test in the second, breaking Draper’s serve in the opening game.
A further break meant Cerundolo had the chance to serve for the set at 5-4 up, but Draper resisted and took the next three games to clinch victory on his first match point.
Draper’s win means he will face 38-year-old Djokovic for the first time since he took the first set off the defending champion on his Wimbledon debut in 2021.