Real claimed their 11th European Cup on penalties as Cristiano Ronaldo scored the winner immediately after Juanfran saw his effort hit a post, with the Atletico full-back the only player to miss from the spot in the shootout.
Sergio Ramos' first-half goal was cancelled out by Atleti substitute Yannick Carrasco as the game ended 1-1 following extra-time but it was Real who came out on top.
It meant Zidane, who only took over from the sacked Rafael Benitez in January, has won the Champions League as both a player and manager for the club.
"At the end getting the Champions League is the best along with winning the World Cup," said the Frenchman, a world champion in 1998.
"After all the work we did I am very proud, it has been a phenomenal job.
"I'm happy because I've been part of this great club for a long time now. I feel really very proud."
Champions League final: Real Madrid vs Atletico Madrid player ratings
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Zidane took over with the club in difficulties but managed to finish the season on a high after domestic disappointment, although questions still remain about his long-term future at the Bernabeu.
"The future? We'll see, now let's enjoy this," he added.
"Of course I dreamed. You always have to dream, think positive, be optimistic, and know that things can get done with work and effort.
Zinedine Zidane is given the 'bumps' by his Real players (Getty)
"I'm very happy for everything we've achieved together. It's not easy at all. We've really worked hard, fought it out.
"When you have players of this calibre, with this talent, you can achieve something big like we have tonight.
Ronaldo claimed he had envisaged scoring the winning goal and paid tribute to Zidane.
"I had a vision. I knew I was going to score the winning goal," he told TV station Antena 3.
"I asked [Zidane] to let me take the fifth penalty because I was going to score the winning goal.
"Zidane has done a phenomenal job. He deserves it, he has humility and I'm happy for him."
Ramos, named man of the match, admitted an 11th European Cup ensured a disappointing season had been turned around.
"We had a unique opportunity to make history after a tough season," he told realmadrid.com.
"We've had some ups and downs and the reward is this title, an award for the work and sacrifice."
We've had some ups and downs and the reward is this title, an award for the work and sacrifice
<p>Sergio Ramos</p>
Atleti coach Diego Simeone, who tasted agonising defeat for the second time in three seasons, was gracious in defeat.
"I'd like to congratulate Real Madrid. Once again they were superior to us, this time at penalties," said the Argentinian, obviously devastated by the result.
"The team that wins is always the best team. I don't believe in excuses.
Diego Simeone tries to encourage Atletico fans to boo Real during the shootout (Getty)
"I need to go home and heal my wounds. I've never had it easy but two finals in three years is a magnificent achievement.
"We need to continue working. I need to think things out on my part.
"It puts a lot of years on you. I don't know which defeat hurts the most.
"What really hurts me is to see the hurt of the people who've paid to come here - I feel a bit responsible that I wasn't able to give them what they wanted. That's what really hurts.
"But I feel proud of my players. We had to defeat Barcelona and Bayern (to get to the final). I love my players."
PA
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