Karim Benzema and Cristiano Ronaldo struck in the last three minutes as the Spanish champions snatched a 3-2 victory from a pulsating Group D clash at the Bernabeu Stadium.
The spirit shown in that late fightback and the overall performance - the hosts having almost completely dominated the first hour - swept away the gloom of a dismal start in La Liga.
Mourinho certainly felt the display was more in keeping with the traditions of their revered, nine-times European champion, club.
Mourinho said: "I am no-one in the history of Real Madrid. I have only arrived recently, I have done very few things.
"But as a coach I am entitled to say - even though I wasn't born a Real Madrid fan like many million people - that Real Madrid fans want this.
"They want commitment, they want better quality of play - and me too. Last year we had it and this year we will have it.
"There was low self-esteem and doubts within the team - and the coach - but it was difficult to play better than we did.
"It was a great commitment from the team, from everyone, without exception."
Real have won just one and lost two of their opening four Primera Division matches, but they seemed determined from the outset to put that behind them.
That they failed to score in the first half had much to do with brilliant City goalkeeper Joe Hart, who saved twice from Ronaldo early on. He also denied Angel di Maria, who later shot wide, and Gonzalo Higuain while Sami Khedira volleyed over.
Marcelo twice narrowly missed the target early in the second half before Edin Dzeko changed the complexion of the game by scoring against the run of play after 68 minutes.
Marcelo quickly responded but Real needed to rouse themselves again after an Aleksandar Kolarov free-kick found its way in six minutes from time.
Even after Benzema equalised again, Hart produced another fine save from Ronaldo, but the former Manchester United star did have the final say.
Mourinho said: "I have been told there were some people leaving the stadium when the score was 2-1, but I felt people were very committed.
"I felt if we had lost, people would have left sad only because of the score. People understood the commitment.
"That is Real Madrid. You can lose but you have to fight until the end."
The result yet again got Real off to a winning start in the competition. The club have now won their first home match in Europe's elite tournament on 41 of the last 43 occasions, with the other two being draws.
Mourinho said: "It was a match that was important to win. We have three points and we play the last match at home.
"I think this is a major step to qualifying from a group that is very hard.
"I don't think we can do what we did last season by getting, almost easily, 18 points."
Real face Ajax in their next match and also play Borussia Dortmund in a group comprised of domestic champions.
Mourinho made a bold decision by leaving out defensive linchpin Sergio Ramos after last weekend's defeat at Sevilla, opting instead for Raphael Varane.
Mourinho told the press: "I would like to make clear - because some of you are experts in changing the truth - there are no problems with Sergio Ramos, no disciplinary issues whatsoever.
"This decision was purely on sports grounds."