The former Liverpool defender started the tie as Real were held 1-1.
"It was incredible to make my debut," Alexander-Arnold said afterwards. "Playing for Real Madrid is something everybody dreams of, although we're disappointed not to have won the game.
"Part of the Real Madrid mentality is winning every game. There were lots of positives to take from it, especially in the second half.
"We had more of the ball in the second half, which is what we were lacking in the first. You have to keep the ball as much as possible to make your opponents chase it. We did that better after the break, we forced them to defend far more and got up and around their box."
No-one to blame
On Fede Valverde's saved penalty, he continued: "The penalty is one of those things in football, we can't be blaming anyone. There are aspects we have to work on and I'm sure we'll improve ahead of the next game."
On the welcome he's experienced from Real fans, Alexander-Arnold also said: "It's been unbelievable. I've felt their support from the moment I signed. There are Real Madrid fans all over the world. You see that support in the hotel, in training, on social media... We want to say thank you. We were heavily supported in the stands and it was fantastic."
And on his improving Spanish, he added: "I've been learning Spanish for a couple of months. Once I knew I was going to come here, I wanted to learn to help me settle in. It was important to me. New country, new team, new city... It's important to show you intend to adapt to the culture in the best possible way and that's what I'm doing."