Serving at five games all in set one, Draper netted three backhands and also struck a doublefault to lose serve and trail 5-6 before breaking serve in the following game to reach a tiebreaker.
In the tiebreaker, Kwon was the more aggressive player with his forehand.
The five feet eleven inches tall player struck several forehand winners including a powerful crosscourt forehand winner on set point to close out the set.
Neither player faced a breakpoint in set two as the set reached a second tiebreaker of the day.
In the tiebreaker, Kwon committed five errors to one from Draper as the Briton looked to be the steadier of the two players mid-match.
As well Draper struck two backhand winners in the tiebreaker including a crosscourt return winner on set point to level the match at one set all.
However in set three, Draper seemed to have left some of his energy and consistency in set two as he struck a down the line backhand wide on breakpoint to drop serve in the opening game of the set to trail 0-1.
After striking several forehand winners, Kwon increased his final set lead to 4-2 before Draper held serve to reach 4-3.
Kwon struck two additional forehand winners in the process of holding serve to lead 5-3.
Draper served to stay in the match and was unfortunately for him no longer the aggressive shotmaker he was mid-match.
Kwon struck a backhand, a forehand and a dropshot winner to reach matchpoint at 30-40.
The South Korean, who had lost in qualifying before entering the main draw as a lucky loser when another competitor withdrew, then hit an inside out crosscourt forehand winner to break serve and end the match.
Kwon used a mix of defense and offense to withstand a barrage of extremely powerful groundstrokes and serves from his taller, more powerful opponent before unleashing his own game changing forehands.
Draper, who turned pro in 2018, posted his best career win in the second round of last year's US Open where he defeated the 7th ranked Felix Auger-Aliassime of Canada 6-4, 6-4, 6-4.
Sunday night in round one at an estimated 8:00pm PST USA, Draper takes on the defending champion and top seed Rafael Nadal of Spain at the Australian Open in Melbourne.