Welsh star Bale, the world’s most expensive player, had missed three regulation time chances before he headed home to make it 2-1 in extra-time as Real won the European title for the 10th time.
“The price tag means nothing to me,” said Bale who moved to Real last year from Tottenham in a world record deal.
“I would have come here for a penny if it meant I could play in the big tournaments and win the big titles.
“It’s unbelievable to lift this trophy — it will be a memory which will live with me forever.”
Bale said he never gave up hope that Real would win the match despite needing an injury-time header from Sergio Ramos to take the final into extra-time.
“We have played Atletico four times this season and they are always close games. It was difficult when they went in front, but we kept trying to break them down,” added Bale.
“But we got the goal in injury time. That gave us the momentum.
“We kept playing our game, we kept attacking and we always believed we could win.”
Bale did briefly fear his missed chances may have proved costly.
“I had a feeling it may not be my night, but I told myself to keep persevering and that one will come.
“When I had the chance in extra-time I just told myself ‘don’t miss it’.”
And Bale’s coach Carlo Ancelotti praised his perseverance to once again be in the right place at the right time after also scoring the goal that handed Real the Copa del Rey against Barcelona last month.
“He tried before in the game and his final shot just wasn’t good enough, but at the right moment he was ready to score.
“Not just today but all season he has been a very important player for us and I am sure that he will be even better next year.”
Ancelotti now becomes only the second man after Bob Paisley to win the competition three times after also achieving the feat with AC Milan in 2003 and 2007.
The Italian, though, admitted that he feared the worst as time was running out towards the end of normal time.
“The hardest thing was getting the equaliser. We suffered a lot in trying to get it because Atletico didn’t give us space and they defended very well.
“We tried with all our energy and eventually the game changed. The goal we scored gave us a lot of strength to keep going.”
Such was Real’s relief at winning their 10th European crown after a 12-year wait that captain Iker Casillas said it meant more than winning the World Cup with Spain four years ago.
“The 10th is more than the World Cup. It has been a long time since we won the Champions League. Ten is a round number,” he said.
“It has not been easy, but we deserved it for how we have played in the tournament this year. We had our bit of luck that we lacked before to equalise and then Bale’s goal killed them.”
The Welshman’s goal was followed by two more in the second half of extra time, with Ronaldo sealing the victory from the penalty spot.