Cristiano Ronaldo scored a 90th-minute winner to give Real a winning start in Champions League Group D, after they had twice fallen behind on a nervy night at the Bernabeu.
"The most important thing for me is to be proud of the team regardless of the result," a pumped-up Mourinho told a news conference, having slid to the ground on his knees while punching the air in celebration at Ronaldo's winner.
"I keep saying the same thing to the players. We can lose matches like tonight, it is not a problem, but fighting like animals like we did to the very end."
Real have made a stuttering start to the defence of their La Liga title, earning only four points from a possible 12 so far.
Their most recent defeat, a 1-0 reverse at Sevilla, prompted a scathing attack by Mourinho on the commitment of some of his players which was still causing reverberations in the Spanish capital ahead of Tuesday's game.
"We cannot lose the way we did in Getafe or Sevilla," said the former Chelsea and Inter Milan boss, who dismissed talk of internal problems in the squad.
"Today we were more aggressive, more physical, we dominated, and we eventually won. We conceded two goals but we kept fighting. We equalised but we kept on going.
"I am no-one in the history of Real Madrid but I have the right to say Madrid fans want to see this. We can play better football, but it was difficult today.
"We were low on confidence, lacking self-belief, but the important thing was the commitment of the players. This is the DNA of last year's team. It is an important step in a difficult group."
Edin Dzeko and then Aleksandar Kolarov had put visiting City within touching distance of a famous victory before Karim Benzema and Ronaldo struck late on for Real.
"It's difficult, not because we lost but because we were beating a team like Real Madrid 2-1 with four minutes to go," City coach Roberto Mancini said.
"We must improve because we go too deep sometimes. We allow opponents to play to the edge of our area. We made this mistake."
Mancini bristled when asked if the English champions, who are competing for only the second season in the Champions League, had perhaps been guilty of naivety in the closing stages of the game.
"We can't continue to talk about experience. We have players who play at international level. We need to improve and should be disappointed with this result," he said.
"We scored two goals which was good, our only mistake was going too deep."
