The 71-year old says his players' "legs went to water" after the Portuguese received a red card during Tuesday evening's Champions League defeat to Real Madrid
The Red Devils held a 2-1 aggregate lead over the Spaniards in the last-16 second-leg clash at Old Trafford before the Portuguese was red carded for a high challenge on Alvaro Arbeloa in the 56th minute.
The visitors subsequently scored two goals in quick succession to put themselves 2-1 ahead on the night, and held on to seal passage to the last eight courtesy of a 3-2 aggregate victory.
And Sir Alex has claimed that his players' disbelief at the "astonishing" decision to give Nani his marching orders changed the game.
"You just couldn't believe it [the red card]. Quite astonishing. I think, in general, we know it was the wrong decision. I think, after looking at the game again a few times, the players were shocked,” he told MUTV.
"They couldn't recover from it until we lost the second goal and then they galvanised themselves and were fantastic in the last 15 minutes of the match.
"They were absolutely brilliant and we could've scored five goals. It was a fantastic performance by the players with 10 men but, in that period after the sending off, they played in shock.
"Knowing we'd had a bad decision against us again, we couldn't get going for 10 minutes. Their legs went to water. That cost us. It really did.
"If it was a justified sending off, in our eyes, it'd have been okay because it was an absolutely brilliant performance tactically. They never made a chance so I think we'd have got through the game."
The 71-year-old took the controversial decision to omit Wayne Rooney from his starting line-up against Los Blancos but believes the risk paid off in the form of a tactically "perfect" performance.
"Our analysis of Real was perfect with exactly all the qualities all the players have. We made sure we were going to be a problem for them," he added.
"Playing Ryan Giggs in off the right-hand side gave the left-back a big problem, whether he should push in or stick and protect the area because Rafael was hunting that space all the time. And Nani's performance in wide left against the boy [Alvaro] Arbeloa was terrific."