Giggs, 39, took part in the 1,000th first class game of his career against Real and in an intriguing interview with his former United team-mate Gary Neville in The Mail on Sunday, Giggs said he was staggered when Nani was sent off.
"I've never, ever experienced a shock like it on a football pitch because I just didn't expect it," he said of the game-changing moment. "And I've never seen a stadium in shock like that."
United were leading 1-0 and 2-1 on aggregate when, in the 56th minute, Turkish referee Cuneyt Cakir gave the Portuguese winger a red card for a high challenge on Alvaro Arbeloa.
The incident had a huge effect with Real coming back to win 2-1 and qualify for the quarter-finals 3-2 on aggregate.
"I'd initially gone over to tell [the referee] Nani was looking at the ball," Giggs told Neville, with whom he played alongside at United for 19 years.
"I've gone back to my position on the edge of the box, because they've got a free-kick.
"I was actually looking at the referee when he did it [showed the red card]. I've never, ever experienced a shock like it on a football pitch because I just didn't expect it."
Sky Sports analyst Neville, who came through the youth ranks at United at the same time as Giggs, said he had been going to Old Trafford since he was five and had never experienced astonishment like it.
Giggs agreed saying: "It was, but the only thing I could think of was 'Right lets try and stay comnpact. Let's try and ride it out.'"
Madrid boss Jose Mourinho brought on Luka Modric after the dismissal and the Croatian equalised with a superb curling shot in the 66th minute before former United hero Cristiano Ronaldo scored what proved to be the winner three minutes later.
Giggs said Modric's excellence, as well as Nani's dismissal helped turn the tie.
"You can't really say anything about the goal; it was brilliant by Luka Modric.
"There's not a lot you can do, going down to 10 men. Carra [Michael Carrick] was in the back four and he had to come out to Modric, close him down, whereas if we'd had 11, he would have been there probably.
"Take nothing away from Modric, it was a great goal."
Giggs said that despite all the hype surrounding the occasion, playing in his 1,000th match did not mean all that much to him.
"I don't want to sound dismissive, that it didn't matter, but no, it didn't. When I retire I will look back at it and I am really proud of getting to 1,000 games but... I just wanted it over with. That doesn't really matter to me, that stuff."
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