Carlo Ancelotti's men trailed Atletico Madrid 1-0 with less than three minutes of injury time remaining, before Sergio Ramos headed in an equaliser from a Luka Modric corner.
Real stormed home in extra-time, as Bale, Marcelo and Cristiano Ronaldo all struck.
Welshman Bale headed home the goal to put Real ahead, which broke Atletico's resolve, and the former Tottenham man said it was a dream result.
"This is for all the fans who have supported us," Bale told the club's website.
"It was a tough match and I'm very happy.
"It's a dream come true, it's the first time I have played in a Champions League final and it was amazing."
Modric, a fellow Spurs defector, hailed Ramos' finish in injury time, and said he and his team-mates were confident of overrunning their city rivals in the extra half-hour.
"We made history tonight and we're very happy," the Croatian said.
"In the side we have match-winners with a lot of quality and we never thought the game was over.
"The stunner that Ramos scored was incredible.
"For us it was key, as in extra time we knew that we'd win as they were very tired."
