Asencio stepped in after Eder Militao suffered a serious knee injury during Saturday's clash.
He told Real Madrid TV, "Since I was little, it's every kid's dream. Being able to watch them on TV all my life and now making this dream come true... I can't find the word to describe it."
He continued: "I entered the field, it didn't take me long to assimilate the situation and I saw Jude's movement, it's very easy to play with them, they are the best in the world. The hug they gave me? In that few words come out at this moment, but it is the demonstration of what the team is, that it is a family and when I train with them. I will feel this gesture in my heart and in my head for the rest of my life."
Asencio added: "I define myself as a player who likes to defend, who shows a lot of character on and off the pitch and likes to help his teammates, serious and energetic and always attentive to his teammates."