The Red Devils led 1-0 early in the second half after Sergio Ramos' own-goal but Nani was then contentiously sent off, giving Real a one-man advantage, and the eventual win following goals from Luka Modric and Cristiano Ronaldo.
The 2-1 triumph at Old Trafford gave Madrid a 3-2 aggregate victory but star midfielder Alonso felt that may not have been the case if Turkish referee Cuneyt Cakir had not brandished the red card.
"It was a difficult decision to take from the referee, but we have seen many games when being against 10 it gets harder. But we have probably produced our best minutes after the sending off," Xabi told Sky Sports.
"For 15 or 20 minutes we have played really well, really patient, and we have scored the two goals. After that we had to push really hard and it was difficult for us not to concede another goal. Our goalkeeper (Diego Lopez) was fantastic."
In addition, Alonso was thrilled with the recent results of the Real side who took down arch rivals Barcelona twice in a week and also progressed to the quarter-finals of the Champions League.
He added: "We knew that it was a key week for us, a make or break week for us.
"It has gone fantastic so far and now we have to look forward. We have two weeks to enjoy watching the other games and wait for the next draw."