Bale, though, insisted that his growing status in world soccer and influence on the Real side would not lead to greater competition between himself and Ronaldo, who scored the fourth goal against Atletico from the penalty spot.
"We work well on the pitch together," Bale said. "We are enjoying playing our football, the whole team. Anybody will tell you no matter who scores the goals we are all happy to win trophies."
Bale's first season in Spain has justified Real's decision to spend a world record 85.2 million fee on him. Tottenham did not make any extra profit from the Champions League success, as the transfer did not include add-ons.
But Spurs did net a multimillion-pound windfall from Luka Modric's final appearance. Real paid Spurs an initial 27.8 million for the Croatian, rising by 5.5 million based on appearances and success, including reaching the Champions League final.
Bale was inundated with messages of congratulations after helping Real win the club's 10th European Cup, with former Liverpool manager Kenny Dalglish believed to be one of the first people to get in touch.
"To win the Champions League in my first season is amazing," Bale said.
"It's the reason I joined Real Madrid to play in competitions like this and be able to win them. For me it is a dream come true and one I'm happy to do."