Carvajal scored in the 2-0 win at Wembley.
He said: "I don't know how to describe it because it's immense joy and it was going to be a very difficult game. In the first half we managed to survive because Dortmund were superior, but we were going to have our moment and in the end we got number 15.
"I would have preferred to win more comfortably. Dortmund were here on merit and in the first half we didn't deserve to go in level. We had to dig in, but that's football.
"This year I was going up for corners and for me it's all about determination. In one I went in a little high and for the second one I headed it in.
"I'm delighted. Especially to win this new European Cup for the club. On a personal level, scoring a goal in a Champions League final is an incredible feeling. Even when the game is deadlocked, it's an incredible feeling. It's been a dream day for me.
"I'm very pleased to be joining this very select club of five players who have won six European Cups. Every time we win one, it's going to be more difficult for them to take it away from us because it's so hard to win a European Cup. I'm really happy, especially for being able to enjoy all six as a starter with my lifelong club. I'm buzzing right now.
"The goal is proof of confidence. When the referee blew the final whistle, I was thrilled to know that my children and my wife were in the stadium. Seeing how I arrived as a kid and here I am, with two children and 21 years defending the shirt of this club and making history with the team. It's fantastic and we're going to enjoy it and celebrate, we deserve it.
"I ran out to celebrate. Inside I felt like there was a kind of scoring rage, a feeling of 'here I am'. It's not all about height, after all, going into the clashes and jumping in with determination often brings you success."