Bale's goal proved decisive as Real eventually over-ran Atleti 4-1 in extra-time.
When asked what it felt like to lift the trophy, Bale told UEFA.com: "A dream come true, literally. When you're a little boy you dream of lifting the Champions League, and to lift the 10th one for Real Madrid is special. It's one that will go down in history.
"It's an amazing feeling to win the Champions League. And yes, it was a hard fought game, they made it very difficult for us, but yes, we showed our character that we've got in the team and eventually we were able to win.
"We always were praying that we'd get a goal, and it was obviously great that we did. We left it late, but it made it that extra bit more special.
"It was a difficult extra time, everybody was tired, but we showed our character again, and we were able to lift ourselves for the last 30 minutes. We got the momentum and got a few goals to win the trophy."
Real boss Carlo Ancelotti commended the attitude of Bale.
His persistence impressed Ancelotti, who said: "It was very important because he had tried before, but he was unlucky in front of goal.
"But he turned up when he was needed."
It is the second time that Bale has scored the winner in a cup final in his first season with Real, with his solo goal against Barcelona in the Copa del Rey earning rave reviews.
Bale's strikes in showpiece encounters will go a long way to silencing the critics that doubted his ability to live up to his world record £85.3million price tag, and Ancelotti is confident the former Southampton and Tottenham man is capable of improvement.
"All season Gareth has been good and he will be better next year," Ancelotti added.
"We're very proud of having won this for all Madrid fans. Happiness is giving happiness to those who always follow us. The fans are very happy and that makes us happy.
"I have had a fantastic group of players who have always been professional and they showed that in this match too. Not only those who were delighted to be playing, but also those who were not playing.
"We've worked hard this season to achieve this."
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