The Wales international, signed in a record record £86million deal from Tottenham last summer, put Real on course for 'La Decima' - their 10th European crown - when he headed home from close range in the 110th minute.
Atletico had been seconds away from glory in Lisbon after Diego Godin's opener in the 36th minute, only for Sergio Ramos to power home a header to take the game into extra time.
Bale's goal knocked the stuffing out of an Atletico side who had given everything, with Marcelo and Cristiano Ronaldo's penalty adding further gloss to the scoreline.
He told Sky Sports: "This is what every footballer dreams of, winning the Champions League, it doesn't come any bigger than that in club football.
"It was an unbelievable feeling, it will be a memory forever.
"The most important thing for me is we worked hard as a team tonight and won the trophy and the 10th title for the club.
"The price-tag means nothing. If I came here for a penny, I came to the club to play in big competitions and win trophies like we have this season.
"The move has paid off now and we want to win more now."
Bale, who had missed several chances before his crucial goal, paid tribute to Atletico, who claimed the La Liga crown this season ahead of Real and Barcelona.
He added: "It was a close game, especially in normal time. We've played against Atletico four times this season and every game is very close and very difficult.
"When they do get a goal ahead it's hard to break them down but luckily we managed to get a goal in injury time and that gave us the extra momentum.
"I think we've been showing that when we play our own football we win football games. It was just keep attacking, keep doing what we do best, and we all believed in that.
"I have learned from past experiences to keep going, to forget about the chances I've missed and keep persevering and eventually one will come. Thankfully I got a decisive goal and it gave us a lot of momentum going into the final 10 minutes."
And when asked what was going through his head as he went to head home after Thibaut Courtois had saved Angel di Maria's initial shot, he said: "Do not miss, literally, do not miss.
"It was great to see it go into the back of the net and the celebrations from the crowd meant everything."