In one hour and eight minutes, Draper defeated the opposition 7-5 6-2 to win the biggest title of his career in Vienna on Sunday.
Before Draper made the pivotal break in the eleventh game, a close first set went to serve.
The British player won the second set after winning five straight games.
On Wednesday, he will now play American fifth seed Taylor Fritz in the second round.
The 22-year-old left-hander, who advanced to his first Grand Slam semifinal at the US Open last month, climbed to 15th place in the global rankings on Monday, a career high.
Carlos Alcaraz of Pain defeated Nicolas Jarry of Chile 7-5 6-1 to advance to the third round.
In the round of 16, the second seed, who received a bye in the first round, will play either American Marcos Giron or Frenchman Ugo Humbert.
Alex de Minaur of Australia defeated Mariano Navone of Argentina 7-5 6-1 in the opening round to increase his prospects of making it to the season-ending ATP Finals.
With a run to the Paris quarterfinals, De Minaur will overtake Andrey Rublev for the final qualifying spot. He will then face Miomir Kecmanovic of Serbia.
