Real are the first to win back-to-back titles since the competition morphed into the Champions League.
Captain Sergio Ramos had called the match a "date with history" and his team responded by creating new chapters in the annals of European football's biggest club competition.
Cristiano Ronaldo scored twice, taking his career tally to an extraordinary 105 Champions League goals, while he is also the first player to find the net in three Champions League finals. His first-half effort was his club's 500th in the competition. Another record.
In his 18 months as Real manager, Zinedine Zidane has now won five titles, including the Champions League twice.
The Frenchman described his team's season as "spectacular."
"They made history, for sure," he told reporters. "Nobody has done it, and now we have. Today is a historic day for all Madridistas, for the players, for me, for all our family."
Although it was Real who opened the scoring through four-time Ballon d'Or winner Ronaldo, the first half belonged to Juventus and, specifically, Mario Mandzukic, whose stunning overhead strike leveled the match. It was a goal which will be remembered as one of the most spectacular in Champions League history.
But treble-chasing Juve's hopes were killed off after the break as Real rose to the occasion to complete their first league and cup double in 59 years and a third European Cup in four years.
Casemiro's deflected long-range strike put the La Liga champions ahead once again, while Ronaldo and substitute Marco Asensio made sure of the victory.
Juventus manager Massimiliano Allegri said his team played "beautifully" in the first half, but admitted his opponents "pressed the accelerator" in the second half.
