Khachanov, seeded 27, fired 30 aces and a total of 63 winners past Wimbledon runner-up Kyrgios in a big-serving contest.
The Russian will face fifth seed Casper Ruud (23) of Norway for a place in Sunday's final.
"I did it, guys. Finally, you are showing me some love," Khachanov told the mostly pro-Kyrgios crowd on Arthur Ashe Stadium.
"It was a crazy match I was expecting it would be like this. I'm ready to run, to fight to play five sets. We played almost four hours and that's the only way to beat Nick."
Khachanov added he has "nothing to lose" when he faces French Open runner-up Ruud on Friday.
"I would like to win it," he said. But as deep as you go the expectations rise up. I did the step forward, I made my first semi-final and I think I have nothing to lose.
"I just want to go for it and be ready for the next match and hopefully it will be a good one."
Heartbreak for Kyrgios
Kyrgios admitted he was "devastated" after suffering what he described as a "heartbreaking" defeat in three hours and 39 minutes in New York.
"I'm obviously devastated. But all credit to Karen. He's a fighter. He's a warrior," said Kyrgios, who had knocked out defending champion and world number one Daniil Medvedev (26) in the last 16.
"I'm just devastated. Just feel like it was either winning it all or nothing at all, to be honest.
"I feel like I've just failed at this event right now. That's what it feels like."
Kyrgios is scheduled to play in Tokyo at the start of October but he admitted he was finding it hard to summon enthusiasm to return to the grind of the tour.
"Honestly I don't even really care about any other tournament. Like, I feel like at the Grand Slams, now having success at a Grand Slam, it's just like no other tournament really matters," he added.
"I think pretty much every other tournament during the year is a waste of time really. You should just run up and show up at a Grand Slam. That's what you're remembered by."
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