The Argentine was named man-of-the-match as Real completed a 4-1 victory after extra-time over rivals Atletico Madrid in Lisbon.
Di Maria, who set up Gareth Bale's goal to put Real ahead for the first time, said his team were deserved champions.
"I feel an incredible sense of relief... it's hard to explain," he told TVE.
"It's so hard to explain what we've achieved. We deserve to be champions. Now we'll keep on celebrating, the team will enjoy this and keep working hard."
Diego Godin put La Liga champions Atletico ahead in the first half before Sergio Ramos' dramatic 93rd-minute header sent the final into extra-time.
Di Maria's dazzling run and shot, kept out wonderfully by Thibaut Courtois, led to Bale heading in before Marcelo and Cristiano Ronaldo scored the sealers.
Delivering Real a record 10th European crown and first since 2002 delighted Di Maria.
"It's a massive joy to win my first Champions League title and on top of that, 'La Decima' for the club," he said.
"I'm really happy, because this is the sort of thing you always dream of as a kid. I'm happy with how much hard work and effort the team have put into this all year.
"We lost out in La Liga in the end and we also lost out on a treble, which is the only shame."
Marcelo was similarly thrilled with Real's win.
"It's a huge joy we’ve been waiting for too long to achieve," he said.
"But we know how hard we’ve always fought, how much we've sacrificed and we've given everything we've got out there on the pitch, to bring joy to our fans."
