The Portuguese winger was shown a straight red by Cuneyt Cakir after a high challenge on Alvaro Arbeloa with the Red Devils 1-0 up on the night, but goals from Luka Modric and Cristiano Ronaldo sent los Blancos through 3-2 on aggregate.
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"I'd initially gone over to tell [Cakir] that Nani was looking at the ball," Giggs told the Daily Mail.
"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. And I've never seen a stadium in shock like that.
"I probably haven't felt that disappointed for a long, long time. The manager always says about games in Europe: 'Be careful because the roof can fall in.' And it did, but not in a way in which you can really blame the players, tactically or some of the performances. It was silent and everyone was gutted. It was shock really."
With a crucial FA Cup quarter-final looming against Chelsea on Sunday afternoon, Giggs has been helping some of the younger squad members through the pain of the Madrid defeat - and insists the Red Devils can move on quickly from the result.
"As an experienced player, you've got something to fall back on," he continued. "You've known this before. A lot of the younger players may not have, but in five days you've got to quickly turn it around to another massive game, a cup game [against Chelsea] when anything could happen.
"For young players, there might be that little doubt in their mind that we've gone out of the Champions League: 'Did I play well or not?' But there wasn't one lad really that didn't play well. It was a good performance. Everything was spot-on, up until the sending-off of Nani, obviously. So there's a lot to be proud of."