Modric spent much of his first season in Madrid after arriving from Tottenham Hotspur in 2012 battling for a place in the first eleven under Real's former coach Jose Mourinho, having to adapt to a less preferred deeper role.
The 28-year old showed at the Stadium of Light how much he has flourished, dishing out passes like confetti and being at the heart of every Real attack before delivering a perfect corner for Sergio Ramos to head in an added time equaliser.
Modric continued to play at full throttle in extra time as Atletico wilted, with several players struggling with cramp as Real turned on the heat to win "La Decima" - their long-awaited 10th title in Europe's premier club competition.
"We knew after we equalised they stood no chance in extra time because some of their players had clearly run out of steam," Modric, draped in a Croatian flag, told reporters.
"This is a dream come true for me because it tops everything I have worked for throughout my career and I am really at a loss for words to describe how overjoyed I am.
"This will give us a tremendous incentive to try to win La Liga next season and also to defend the Champions League title, having taken a huge stone off our backs by clinching this one.
"We were almost obsessed with the task because the expectations at this club are always high and it's been a long wait."
Modric, who started as an attacking midfielder at his first club Dinamo Zagreb, gradually morphed into a more versatile player deployed across the engine room at Spurs before his move to Real, where he initially struggled to fit in.
Having shone this term in a playmaking role under Carlo Ancelotti after the Italian took over from Mourinho, Modric credited the two managers with his improvement.
"They are both outstanding coaches and great personalities," he said. "Being signed by Real was the happiest moment of my life up to this point and the reason I am here today celebrating the biggest accomplishment of my career is because I never stopped believing in myself.
"Likewise, we never lost faith tonight that we would draw level after falling behind and the equaliser was just reward for our efforts. Atletico were worthy opponents in every aspect but we are team full of natural born winners who have performed many times on the big stage.
"It's been a long season but this feeling of pure joy will ease the fatigue ahead of the World Cup in Brazil, which I am very much looking forward to after a few days' rest."
Croatia, the 1998 World Cup semi-finalists, are in Group A with Brazil, Cameroon and Mexico.