Diego Godin's first-half header had looked like denying the Welshman a first European club winner's medal, but Sergio Ramos nodded home a corner before the full-time whistle and Bale scored one of Real's three extra-time goals as they ended a 12-year quest for 'La Decima'.
The world's most expensive player suggests that Ramos's equaliser was crucial to Los Blancos's success.
He told reporters: "We played Atletico four times already this season, every one was very close and difficult, when they get ahead they are difficult to break down. Once we got the goal, we had the momentum into extra time.
"When we play our game, we win football games. We knew if we kept attacking, we would win."
Bale scored the winning goal in Real's other cup success this season, the 2-1 Copa Del Rey triumph over Barcelona.