Madrid's desire to claim a record 10th European Cup crown was floundering badly prior to Mourinho's arrival in 2010, with the Spanish giants suffering six successive exits at the last-16 stage.
However, since the Portuguese took charge Madrid have re-established themselves as a force to be reckoned with on the continent with three successive semi-final appearances.
The first two ended in disappointment with defeats to Barcelona and Bayern Munich, but Mourinho hopes it will be third time lucky when they take on Borussia Dortmund and that his team can then go on to lift the trophy he believes they deserve to win.
"We have a way of playing which is well-known, we've been on the brink for three years and we have s style of the play that guarantees quality," he said.
"I would like football to do justice to our players because with a few exceptions, none of them have won this competition so far and we've never won it as a team, and they deserve to do so."
"We've fallen in two semi-finals, one is best forgotten (against Barcelona) and the other on penalties (against Bayern) in a very tough manner.
"Hopefully football will now be fair with my players."
He added: "At this club to finish second is not enough, nor is it to reach the semi-finals.
"We need to give our maximum to reach the final."
Madrid met Dortmund in the 1998 Champions League semi-finals and knocked the then holders out 2-0 aggregate defeat before going on to lift the trophy themselves.
Dortmund have had the better of the meetings between two in the season's competition though, winning 2-1 at home in the group stages before drawing 2-2 at the Bernabeu in a match which saw Madrid net an 89th-minute equaliser through Mesut Ozil.
Mourinho concedes his side will need to defend better than in those games, saying: "All the goals they scored against us in the group stages were our own mistakes. We could have avoided them.
"In this tie we will have to be much better focused defensively. We will be more solid. If they score against us they need to be fantastic goals or because we haven't been able to do anything about them."
Mourinho, whose side could lose their Primera Division title to Barcelona this weekend if results go against them, opted to reveal at Tuesday's pre-match press conference what his side will be for the first leg, with the only question mark being winger Angel Di Maria who is set to travel to Germany on Wednesday following the birth of his child on Monday.
Mourinho has again preferred Diego Lopez to Iker Casillas in goal, while Sergio Ramos will line up at right-back, Fabio Coentrao will play at left-back with Marcelo out injured and Gonzalo Higuain will start up front ahead of Karim Benzema.
In announcing his team, Mourinho added: "We hope that Di Maria plays, who has had a happy day with the birth of his daughter but a tough day emotionally."
Regarding the clash with Dortmund, Mourinho said: "The tie is 50-50. Both teams have the same chance of going through.
"(Jurgen) Klopp is a great coach. I don't know him very well but he's genuine, honest.
"When I read that he doesn't concern himself at all with Real Madrid and is only concerned about his team, I cannot expect anything else than a Borussia who will come out and give it everything and be attacking."