Atletico seemed set for their first ever European crown as they led deep into stoppage time thanks to Diego Godin's first-half header, but Sergio Ramos took the game into extra time with a towering header in the 93rd minute yesterday.
And for the second time this season having already scored the winner to hand Real the Copa del Rey against Barcelona last month, Bale scored the crucial goal when he headed into an empty net after Thibaut Courtois had parried Angel di Maria's effort.
"It is the most important goal I have scored," said Ramos.
"It is an incredible feeling. We kept fighting until the end for the fans. It is not my goal, but one for all Madrid fans, my family and all those that have supported us until the end. We deserved it after so many years."
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Real coach Carlo Ancelotti, meanwhile, becomes the first man since Bob Paisley to win the competition three times.
"I am very happy for this because we have won a trophy that for Real Madrid is very important. We have worked hard all year for this."
Atletico captain Gabi admitted that his side had been well-beaten in extra time after suffering the body blow of Ramos' equaliser.
"They managed to equalise and that is football," he told Television Espanola.
"After the extra time they were deserving winners. We have competed for a year like animals and we have gone as far as we could."
"This match doesn't deserve a single tear," said the Argentine.