United took the lead during an eagerly-anticipated last-16 encounter at Old Trafford, with an own goal from Sergio Ramos edging them in front on aggregate.
The two-legged tie turned, though, on the 56th-minute dismissal of Nani, with the Portuguese winger given his marching orders after catching Alvaro Arbeloa with a high boot.
Alonso concedes that being handed a numerical advantage played into the visitors' hands, with Luka Modric and Cristiano Ronaldo netting to see the Spanish giants through to the quarter-finals.
The Real midfielder told Sky Sports: "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.
"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 was fantastic."
Real have now beaten Barcelona twice and Manchester United in their last three fixtures in all competitions and Alonso hopes they can go from strength to strength after coming through a testing period.
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."