The 21-year-old, ranked world No.123, recorded an impressive 6-4 6-2 3-6 6-3 victory over American world No.89 Michael Mmoh.
Draper will next face Russian eighth-seed Andrey Rublev or Frenchman Arthur Rinderknech as he bids to reach the quarter-finals at Flushing Meadows.
‘It was a tough match,’ Draper said in his on-court interview after beating Mmoh in New York.
‘He’s an incredible player and he’s had a great year so I knew it would be a real challenge today.
‘I was proud of the way I came through. It was tough conditions at times, but amazing to play on this court (Grandstand) in front of everyone so thank you.
‘Last year here was really difficult. I was in the third round and I injured myself. I didn’t want that to happen again today.
‘It was a long match (today) and to come through after a tough year for me, I am very happy. What a place to do it (reach the fourth round of a Grand Slam), there is amazing energy in New York.
‘So many people come out and I appreciate the support. It’s not easy out here. I look forward to playing in the fourth round.’
Asked if he will watch Rublev’s match against world No.73 Rinderknech, with Draper set to face the winner, he added: ‘I’ll watch a little bit! Both of them are incredible players. Anyone who I play is going to be tough.’