"We won a cup which is all the more valuable because of the quality of our adversary and the moment that sticks with me is my embrace from the president (Florentino Perez)," Mourinho told a news conference.
"To feel that people want me, and that I will be here for more than one year, makes me feel more relaxed. My teams are (always) better in the second season."
The former Inter Milan, Chelsea and Porto coach's future at the Bernabeu has been the subject of media speculation despite him being in the first season of a four-year deal.
The pressure to win a trophy for Real in Mourinho's debut campaign eased after Wednesday's 1-0 King's Cup win over runaway La Liga leaders Barcelona.
Mourinho also dismissed media reports linking him with a return next season to Inter, the team he led to a Champions League-Serie A-Italian Cup treble last year.
Real trail the Spanish league leaders by eight points with six matches remaining and visit third-placed Valencia on Saturday before doing battle with Barca again in a Champions League semi-final first leg on Wednesday.
Injuries will keep out Emmanuel Adebayor, Angel Di Maria and Sami Khedira for the game against Valencia at the Mestalla - the venue for Real's win in the King's Cup.
Germany midfielder Khedira's leg injury is likely to rule him out of the midweek match with Barca as well.
"He won't play tomorrow or next Wednesday, almost certainly," said Mourinho. "But we hope he'll be back as soon as possible."
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