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.
“I’ve been watching him since I was a young boy and in my opinion he’s the greatest player of all time,” said Draper.
“When you come up against him you know he’s going to be there mentally, you know he’s going to make it an incredibly tough match and play some crazy tennis. I’m going to have to be ready for that.”
The victories for Norrie and Draper came a day on from fellow Briton Emma Raducanu being well beaten by American Amanda Anisimova, winning only two games and being ousted in 52 minutes.
Norrie is the 27th seed in the men’s draw but played above that status as he earned three breaks en route to victory over De Minaur.
His next opponent will be Australian qualifier Rinky Hijikata, who beat Kazakhstan’s 10th seed Alexander Bublik in three sets.
