Carlo Ancelotti's men needed a 93rd-minute equaliser from Sergio Ramos to force extra-time in Lisbon on Saturday, and they went on to claim a dominant 4-1 win.
Gareth Bale, Marcelo and Ronaldo struck in the additional period to help Real to a record 10th European crown and first since 2002.
Despite needing the last-gasp equaliser to stay in the game, Ronaldo said Real were deserved victors.
"We deserved to win because we played better. They didn't have any chances and we were better," Ronaldo said.
"That's football, there are matches you can win in the final minutes and that happened today. Atletico should be congratulated for their great season."
Real defender Pepe, who was on the bench throughout, agreed with Ronaldo and praised the club's fans.
"This competition is very important for us. We followed the tradition of Madrid in fighting to the end," Pepe said.
"I think we were better for the whole game. They scored from a dead ball through our own mistake, but we fought and knew that together with the fans we could win the Decima."
