The annual meeting of the Champions League and Europa League winners proved an entertaining encounter, with the Red Devils impressing at times and being outclassed on many others.
Romelu Lukaku's late goal gave United hope after Casemiro and Isco put Real Madrid into what ultimately proved an unassailable lead, with Zinedine Zidane's men triumphing 2-1 and wrapping up back-to-back Super Cups.
More attackers considering Chelsea exit amid problems with Tuchel - report
Mourinho was annoyed that Casemiro was not called offside for the first-half opener, but the United boss found it hard to argue that his former club did not deserve the lead at that point.
"There was a goal that was offside," he said. "If not, we would be talking about 0-0 at half-time and a different game.
"But the domination is the consequence of quality of a team that is superior, that has great quality players in midfield that control the game, that hold the ball, that keep possession.
"They touch the ball well, they pass it very well. It is a consequence of that."
Unlike many occasions, though, Mourinho did not get caught up in that frustration - instead he focused on the positives ahead of Sunday's Premier League opener against West Ham.
"I am happy with the performance or many things of the performance," he said. "We don't have in the Premier League Real Madrid. In spite we have top teams in the Premier League, but they are different teams - teams with different qualities.
"The midfield players of Real Madrid there are only one, there are no replicas.
"No replicas of (Luka) Modric, Toni Kroos, Isco, no replicas of Casimero, so they are really solid and the way they move the ball makes it difficult for the opponents.
"In the second half we found a way to create them difficulties and to give them the kind of football they don't like and don't dominate.
"I am really happy with our reaction because at 2-0, the next minute we have a double big chance with just Navas in front.
"And after the 2-1, we had Marcus Rashford face-to-face with the keeper and we had a fight until the last seconds and to do that I think is great because I always say that the Champions League is one thing and the Europa League is another thing.
"The Champions League winner obviously has a different potential than the Europa League winner, so for us to come here and to do the positive things we did, I think we have to go with a good feeling and we have to start the Premier League on Sunday with a positive feeling and great experience.
"And going to the Champions League in September, who knows? If we get Real Madrid again, if we don't we could get another of the top teams in Europe - one of the champions of the best leagues - and I think this Super Cup was a really good experience for us."
Mourinho praised his "amazing group" and full debutant Nemanja Matic, having already made his delight at the United support known to them after the match.
The Portuguese made one supporter's day in particular, giving them the runners-up medal he had just received.
"Look, sometimes when I win I don't keep the medals so imagine when I lose?" he said to laughter in the post-match press conference.
For me, the medal would go to some place in my house and for that kid it is the moon.
"For that kid, it is for sure something that he is going to keep and to remember. It was a little kid with a Man United shirt – I have no idea where he comes from.
"I think for the kid it is something he will never forget. For me, medals - I repeat - when I win, they don't mean much. Imagine when I lose."
Real Madrid counterpart Zidane was proud of his players' efforts, saying they "never get tired of winning" having continued their remarkable run of success under his tutelage.
"You cannot play and play beautiful football at the same level for 90 minutes," he said of their testing ending.
"We played at a high quality, we suffered, yes, but we controlled the game. If Gareth Bale scores the third it's over.
"But we were against an opponent who wanted to cause us problems and it's normal if we suffered a bit."
Love Manchester United? Connect with fans at Manchester United Forum on Facebook and manutdforum.org
'He gave everything' - Tuchel delighted with Lukaku
Jose Mourinho: The end? – The Warm-Up