The 21-year-old Brit’s win against the eighth-seeded American atoned for his loss to eighth seed Karen Khachanov last week in Adelaide — with a potential rematch on the cards, if Khachanov can defeat Marc-Andrea Huesler later in the day.
Draper converted on three of five break chances against Paul, who was playing in his first event of the new season.
Broken just once, Draper fired 10 aces and won 88% of his first-serve points in the win.
Draper, who earlier dispatched Lorenzo Sonego 6-4 6-2, reached three ATP Tour quarter-finals last season. The Briton broke serve four times in his opening victory against the Italian.
The Adelaide International 2 is the second of consecutive ATP 250 tournaments staged in the city with Novak Djokovic winning the first instalment on Sunday.