Asensio was front and centre as Madrid sealed a 5-1 aggregate triumph with a comfortable 2-0 victory at the Santiago Bernabeu on Wednesday.
"My two goals in the Spanish Super Cup were great and I will look back on them fondly," the 21-year-old star said.
"The fans show us incredible support and so naturally we want to reward them with titles.
"We were playing at home and had to really turn it on, and we created opportunities. We played the better football over the tie."
Madrid's title defence starts away to Deportivo La Coruna on Sunday.