The rocker disclosed to The Sun that he initially disliked Cash's "invasive" version because it was a song that was intensely personal to him.
He said: "[Producer Rick Rubin] called me to ask how I'd feel if Johnny Cash covered 'Hurt'. I said I'd be very flattered but was given no indication it would actually be recorded.
"Two weeks went by. Then I got a CD in the post. I listened to it and it was very strange. It was this other person inhabiting my most personal song.
"I'd known where I was when I wrote it. I know what I was thinking about. I know how I felt. Hearing it was like someone kissing your girlfriend. It felt invasive."
However, Reznor admitted that he was won over by the cover when he saw the accompanying music video, saying: "What a powerful piece of art. I never got to meet Johnny but I'm happy I contributed the way I did.
"It felt like a warm hug. For anyone who hasn't seen [the video], I highly recommend checking it out. I have goose bumps right now thinking about it."
Reznor added: "That's something that matters to me. It's not so much what other people think but the fact that this guy felt that it was worthy of interpreting."