The Argentine worked tirelessly down the wing to keep his side in the final of Europe's elite club competition after Atletico went 1-0 up after 36 minutes at the Stadium of Light.
In extra time, he had to drop back because of an injured leg and allow substitute Marcelo to surge forward.
But Di Maria finished the game and his efforts were instrumental in Real's 4-1 win.
"I felt a small pain in the back of my leg. I couldn't really run so I decided to stay back and have Marcelo play in front of me," the 26-year-old Argentina international said.
"For me it's fantastic to have won this cup - it's a dream I've had since I was a small child.
"As the group we worked very hard all year long. We lost La Liga... we've made many sacrifices all the year and it's great to win it," he added.
Real faded in the Spanish title race and finished third in the Spanish league behind Atletico and Barcelona.
Di Maria, who played for Benfica before joining Real, said he felt at home back in the Portuguese club's stadium.
"I was happy here and did some good things here that got me the move to Real Madrid and it really makes me happy to win Champions League here too."