After Real's second goal, United threw caution to the wind and finished the game on top, denied an equaliser and possibly more only by the heroics of goalkeeper Diego Lopez.
Mourinho continued: "When we were against 10, we played very well for 10 minutes. We created them new problems with [Mesut] Ozil wide and Modric and Kaka both behind [Gonzalo] Higuain. I think it was a new problem for them that they couldn't solve with 10 men. We deserve credit for that.
"But at the end of the match, we couldn't control the possession, which frustrated me because we had one player more. We should have controlled the match but we couldn't and at that moment we had Diego Lopez.
"To have a fantastic keeper who gives us victory is also part of the game. [David] De Gea was the man of the match in Madrid and today the man of the match was Diego Lopez."
Real midfielder Xabi Alonso added: "It was a difficult decision to take from the referee, but we have seen in many previous games that it can be harder playing against 10 men.
"We produced our best minutes after the sending-off. After that, United pushed really hard, it was difficult not to concede and our goalkeeper was fantastic."
Mourinho and Alonso were speaking to Sky Sports and ITV Sport.