The Swiss, who also upset three-time defending champion Novak Djokovic in the quarter-finals, capped a magnificent fortnight with his 6-3, 6-2, 3-6, 6-3 win over Nadal for his first major title.
"I still think that I'm dreaming. It's a strange feeling," he said.
"I've seen so many finals. I always try to watch the final of Grand Slams because that's where the best players are playing.
"Before today for me, it wasn't (even) a dream. I never expected to play a final. I never expected to win a Grand Slam. And right now I just did it."
He added: "To beat Rafa, even if he was injured, I think I played my best first set during the match. I was ready to play four hours or five to beat Novak in the quarters, to beat Tomas Berdych in the semis."
Wawrinka became the first player since Sergi Bruguera at the 1993 French Open to beat the two top seeds, Nadal and Djokovic.
He said he spoke with his Swedish coach Magnus Norman about playing in a Grand Slam final, and he was surprised how calm he was at the start of the match.
The Swiss got off to a brilliant start, taking his first ever set off Nadal in 12 matches and going a break up in the second before the Spaniard's back problem became apparent.
He went two sets up but lost the third as Nadal threatened a comeback, before stepping up to take the fourth set and the championship.
"Magnus told me it was important not to think about the result but think about the way you want to play, the way you want to win every point," he said.
"I was surprised how well I started the match. In the beginning, he was good, he was fit, he wasn't injured. And I was playing amazing tennis.
"Then there was the second part of the match where I had to stay calm and just to try to stay aggressive because he was injured, but he was still trying a little bit.
"It was not easy. I started to be really nervous because I started to realise that I could win a Grand Slam. But at the end I just came back to the game and focused on what I wanted to do."
Adding spice to his achievement is that Wawrinka has supplanted Federer as the number one ranked Swiss player, a position his close friend has occupied since 2001.
Wawrinka's rise from world number 17 at year-end 2012 to three stems from his monumental fourth round match with Djokovic at last year's Australian Open, which went 12-10 in the fifth set.
He carried this momentum on to his tight defeat in New York, and his redeeming victory over Djokovic in the quarters this week, before Sunday's breakthrough win.