Cristiano Ronaldo scored twice, with further goals from Casemiro and Marco Asensio consigning Mario Mandzukic's blockbuster to a scant corner of European Cup history.
Zidane has not yet served two full seasons as a head coach but has a pair of Champions League crowns to his name, adding to the 2002 title he won as a player at the Santiago Bernabeu.
Madrid also secured a league and cup double for the first time in almost 60 years, leaving Zidane ecstatic in his post-match news conference.
"No one did it [defending the Champions League] and we did it so you can say today is a truly historic day for all Madrid fans, for myself, for the whole [Madrid] family," he said.
"But you know how things are. Next year it's going to be even more difficult and we are going to have to work very hard to be able to win once again.
"Today we can say that we are a very good team and that we deserve to win the league and the Champions League, which was not easy."
The final in Cardiff was finely poised at 1-1 heading into half-time but Juve wilted in the second period as Madrid ran away with it.
"It's been a spectacular year, I couldn't dream for better," Zidane said.
"In the end you're playing against a great team and it's a 50-50 chance but we started with difficulty.
"Second half we were clearly superior, we kept the ball, physically I think we won the game there as well. Thanks to the determination we showed I'm so very happy."