The 21-year-old beat Khachanov at the US Open last summer but lost to the Russian in the first Adelaide tournament last week.
It was a different story on Thursday though, as British number three Draper took control of proceedings and clinched a 6-4 7-6 (3) victory over the third seed, beating a top-20 opponent for the sixth time.
Draper had a minor wobble when he was broken trying to serve out the match at 5-4 in the second set but he recovered impressively to take it in the tie-break.
Draper said in his on-court interview: "Against guys like Karen, you can't give them an inch. I got a bit nervous but that's all part of the game.
"I'm still young, still learning. It was a good match. I did well to impose my game a little bit more today than last week."
The young Londoner lost his first semi-final at Eastbourne last summer and was also beaten in the last four at the Next Gen ATP Finals in November for the best young players.
Cameron Norrie also battled his way into the last four at the ASB Classic in his home city of Auckland.
The British number one, who grew up in New Zealand before moving to London as a teenager, was pegged back by USA's Marcos Giron before coming through 6-1 6-7 (5) 6-2.
Norrie said: "I came out firing and really was enjoying my level and playing well. I made it really tricky for myself and played a loose game to start the second set. He played a better tie-break than me, was more aggressive. Then I came out and played kind of similar to the first set and took it to him."
Second-seed Norrie is the highest-ranked player left following the shock defeat of world number three Casper Ruud against Laslo Djere.