Carlo Ancelotti's side won their first major European trophy in 12 years as they defeated rivals Atletico Madrid 4-1 in Lisbon.
Real were grateful to a stoppage-time equaliser from Sergio Ramos to keep them in the game, and in extra time they took command of the match as goals from Bale, Marcelo and Cristiano Ronaldo secured the win.
"It will be a memory forever and it'll be great to look back on in future years," Bale told reporters.
"We want to win more now. This is what every player dreams of, winning the Champions League."
Real Madrid also won the Copa del Rey this season.