Draper, who turned professional in 2018, recovered from trailing 15-30 in his following service game, striking an ace down the tee to hold serve and reach a tiebreaker at six games all.
Khachanov hit a down the line backhand winner to open the tiebreaker and lead 1-0.
The powerful lefty Draper then struck a crosscourt forehand winner and an ace down the tee, his tenth of the match, to lead 4-2 in the tiebreaker,
Khachanov committed two backhand errors to trail 2-6 and face matchpoint.
Khachanov struck a crosscourt backhand volley winner to reach 6-3.
On his second matchpoint, Draper served and volleyed; hitting a down the line forehand volley winner to close out the match after one hour and forty-five minutes of play.
In Friday's semifinals, Draper will meet the 84th ranked lucky loser Soonwoo Kwon of South Korea.
Draper has not yet played the Australian Open and posted his best career grand slam result at last year's US Open where he reached the third round.