Draper lost to the Russian world number 20 in the first of the two Adelaide tournaments last week but dominated Thursday's quarter-final with his powerful game.
The 21-year-old stumbled to finish off the contest and was broken after leading 30-0 while serving for the match, but made no mistake in the tiebreak to set up a last-four meeting with Kwon Soon-woo.
"Against guys like Karen, you can't give them an inch or they run away with it," Draper said of the 2022 U.S. Open semi-finalist.
"Last game I got a bit nervous but that's all part of the game, I'm still young, I'm still learning. It was a good match. I did well to impose my game on him."
British number one Cameron Norrie also reached the semi-finals at the other ATP warm-up being played this week, beating qualifier Marcos Giron 6-1 6-7(5) 6-2 between the rain showers at the Auckland Open.
Norrie whipped through the first set but was dragged into a baseline dogfight in the second and the American levelled up the contest in a tiebreak of the highest quality.
After his second-round win on Wednesday was completed indoors with no fans because of the weather, Norrie finally got his chance to address the crowd in the city where he was raised.
"It's so good to be back, it's a second home for me, it's great to be playing on this court," the world number 12 said after taking his record to 5-0 for the year.
"I made it difficult for myself in that second set, it's kind of annoying not to get it done in straight sets but credit to Marcos, he's a great player."
While the players in Auckland have had to deal with the rain, Draper was getting accustomed to the Australian summer heat.
"I prefer the cold, I reckon. It's brutal when you're out playing," he said.
"I need to get used to it because the tour follows the sun."
