Draper, the world number 25, defeated 10th-ranked Alex de Minaur of Australia 6-3, 7-5, 6-2 and goes on to face either top seed Jannik Sinner or 2021 champion Daniil Medvedev for a place in Sunday's final.
It's an achievement that a little over a year ago would have seemed unlikely.
Starting 2023 at 38 in the world, Draper then suffered multiple shoulder injuries and missed huge chunks of the season, including most of the clay swing and all of the grass court season.
His ranking slipped out of the top 100, forcing him onto the second-tier Challenger circuit before a confidence-boosting run to the last 16 at the US Open.
Twelve months on, he is on the verge of the top 20 and the biggest match of his career coming up on Friday on the world's biggest tennis stadium.
"Last year was a real turning point for me, when I had my injury setbacks and taking a lot of time off over the summer because of my shoulder injury," said Draper.
"I had to watch all these young, amazing players winning amazing tournaments. But I knew that my time would come. I didn't know when it would be, but hopefully from here I can do a lot of amazing things. I'm very proud of myself."
The 22-year-old left-hander pulled off victory on Wednesday despite taking a medical timeout early in the second set to have his right thigh bandaged.
But he was quick to allay concerns that the injury may be an issue for Friday's semi-final.
"At the end of the day it didn't get any worse, and I felt good towards the end," said the Briton who had lost three times in three meetings with De Minaur before Wednesday.
Draper has made the semi-finals without dropping a set as he continued an impressive summer run which saw him capture his first ATP title in Stuttgart and then defeat Carlos Alcaraz at Queen's Club on the eve of Wimbledon.
'This is a privilege'
In New York, he has won 60 of his 63 service games. He sent down 11 aces in his 40 winners on Wednesday and was rewarded for his all-out attack, forcing De Minaur to fend off 14 of 20 break points he faced.
It was a performance which allowed Draper to put last year's dark times behind him.
"I've had setbacks, I've had times when I've maybe thought to stop or am I cut out for this sport, am I really good enough and all this sort of stuff," he said.
"I kept on believing in myself, kept on working. Those were hard moments. This is kind of not a hard moment compared to that. This is a privilege, and this is an honour to be in this position."
De Minaur admitted he had aggravated his recent hip problem during the match.
"Stopping didn't cross my mind at any stage," said De Minaur who was forced to withdraw from Wimbledon on the eve of his quarter-final with Novak Djokovic due to the hip issue.
"It's not part of DNA. I'm always going to give my best."
Draper enjoys a 1-0 lead over Sinner in the pair's head-to-head although that win at Queen's came three years ago.
He lost to Medvedev on clay in Rome earlier this year.
