Real fell behind to an early Diego Godin goal but fought back, with Sergio Ramos forcing extra time with a last-minute header and Bale later made it 2-1, before Marcelo and Cristiano Ronaldo rounded off the rout.
And, speaking after the game, the Welshman insisted that “it doesn't come any bigger” than winning European football's biggest prize.
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."
Bale also claimed he never felt any doubt that Real would go on to clinch victory, despite Godin handing Diego Simeone's side the lead.
"We've played against Atletico four times already this season and every game is very close and very difficult,” he added. “When they do get a goal ahead it's hard to break them down. Luckily we managed to get a goal in stoppage time and that gave us the momentum.
"We've been showing that all season. When we play our own football we win football games and it was literally just keep attacking, keep doing what we do best and we all believed in that and we all believed if we kept doing that we'd win.
"It was an unbelievable feeling [lifting the trophy] - one you can't describe and yeah, it'll be a memory forever and it'll be great to see it in future years.”
And Bale also laughed off suggestions that his goal paid back his £85m price-tag, adding: "For me, the price tag means nothing. If I came here for a penny I wanted to play well.
“This is why I came to the club, to play in big competitions and win trophies like we have this season. Obviously the move has paid off this year and we want to win more now."