The success brought Madrid their 10th European Cup, with Isco "very proud to go down in the history" of the famous Spanish side.
"I'm very proud to go down in the history of this club. The match was tough, but we are Madrid and we never give up," he told reporters at a press conference.
"We kept fighting until the end and that makes victory taste even sweeter. We're really happy to give the fans so much joy. La Decima is a title that Madrid has been looking for for a long time and we have to celebrate accordingly."
Madrid also won the Copa del Rey this season.