Draper hit back from losing the first set on a tie-break to take the second 4-1, the 20-year-old left-hander producing a stunning array of passing shots against his American opponent.
However, Nakashima broke in the sixth game of the third set to take it 4-2 and reeled off five straight points from 5-2 down in the fourth set tie-break to seal an impressive win.
Nakashima will take on Czech Jiri Lehecka in the final after Lehecka defeated Switzerland's Dominic Stricker in four sets in the first semi-final.
"It was very high quality from both of us the whole match," Nakashima told the ATP Tour's website.
"I was fighting for every point and I was happy to close it out at the end. We have both had a great year and both had great wins against top players.
"I knew it was going to be a tough match, he has a great lefty serve. I knew I had to be quick on my feet. I am happy I played well. I have a lot of confidence in my game right now."
Nakashima, who beat Lehecka in the group stages, added: "I am going to try and forget about our match a couple of days ago. Jiri has been playing great all week, so I am sure it is going to be a tough one."