British quantity 4 Draper wanted just one hour 20 minutes to finish his victory in opposition to world quantity 14 Paul in Adelaide.
Having powered right into a 4-0 lead, the 22-year-old broke for a 3rd time at 5-1 to take the opening set.
Paul, who additionally misplaced to Draper in Adelaide final yr, got here out stronger within the second set and managed to carry serve till the Briton received the decisive break at 5-4 and secured the win in fashion with an excellent passing shot down the road.
If Draper beats Bublik within the semi-finals, he would face American Sebastian Korda or Czech seventh seed Jiri Lehecka within the remaining.
“I am actually impressed with the way in which I got here out right this moment,” stated Draper, who survived two match factors to beat Miomir Kecmanovic on Wednesday.
“I knew it was going to be tough, I performed a extremely lengthy match yesterday. I am actually pleased with the work I am doing, exhibiting my physicality on courtroom.
“Every time I play I am proving myself to be a greater participant than I believed I used to be, so a extremely good match in opposition to an ideal participant like Tommy.”