The Portuguese netted the decisive spot-kick following a 1-1 extra-time draw between Real Madrid and Atletico Madrid at the San Siro.
Sergio Ramos had given Real an early lead but, after Yannick Ferreira-Carrasco equalised, Juanfran missed Atletico’s fourth penalty and Ronaldo stepped up to seal victory.
“I had a vision and saw I was going to score the winning goal, so I asked (Zinedine) Zidane to take the fifth penalty,” said Ronaldo.
“It is a very special moment, and it is well deserved for the players and the fans, who’ve always been there for us.”
Ronaldo has now won the Champions League three times while Real head coach Zidane is only the seventh man to win Europe’s premier club competition both as a player and coach.
“Zidane has done a phenomenal job and he deserves it because he’s a gentleman and is humble, so I’m happy for him,” said Ronaldo.
Wales winger Gareth Bale overcame cramp to score his spot-kick and revelled in an “amazing feeling”.
“It was difficult, in extra-time a lot of people were getting cramp,” he said. “We showed resilience and what we’re made of.
“It’s an amazing feeling, I can’t describe what it feels like. The boys gave everything. The club, the fans have been amazing on this journey – we deserve it.”
Croatia midfielder Luka Modric won the Champions League two years ago with Real and praised his side for fighting on despite the tough conditions at San Siro.
He said: “Thank God we won it again. It was a hard, hard game from the beginning to the end. The weather was difficult tonight, a lot of humidity but we fought, we fought as a team until the end and we made it.
“We are very happy and proud of the team, we did a great job. We showed character at the end – we were calm on the penalties.
“I think we deserved to win. I feel sorry for Atletico but I’m glad for ourselves that we won it again.”
After clinching the trophy for the 11th time in the club’s history, Modric added: “This is the competiton of Real Madrid. This trophy belongs to Real Madrid, that’s why Real Madrid is the best club in the world.
“We suffered a lot, we suffered as a team but in the end we deserved it.”