Zidane was an assistant to Carlo Ancelotti when Madrid won its 10th Champions League trophy in 2014 to end its title drought.
With the crowd in a frenzy, four-time Ballon d'Or victor Ronaldo - who finished the season with 42 goals in all competitions and as the Champions League's top scorer - ended up singing, "Cristiano, Ballon d'Or" and the fans joined in.
With a third Champions League title in four seasons secured, having dethroned Barca for their first La Liga silverware since 2012, there were wild celebrations in Madrid.
"But I'm extremely happy and I'm very grateful to the club for giving me the opportunity to coach these fantastic players".
Madrid became the first team to defend the European Cup successfully since Arrigo Sacchi's AC Milan in 1990 and the first to achieve the feat since the competition relaunched as the Champions League in 1992.
"It's a tremendous joy for the players and for this enormous club", Zidane told beIN Sports Spain.
"My family is happy and I am happy, so yes we will continue what we are doing. This club is the biggest in Champions League history ever, it's great to be a part of and we want to win more".
Bale's injury woes this season - and the impressive displays of Spanish playmaker Isco in his absence - have led to suggestions that the Wales striker could return to the Premier League this summer.
Real Madrid's players are set to be handed record-breaking bonuses to match their historic season. "So yes, just happy to get on for the last few minutes".
Leonardo Bonucci's raking pass was volleyed into the box by Alex Sandro and Gonzalo Higuain kept the ball airborne for Mandzukic to chest it up and send a looping volley over Keylor Navas.
The Croatian forward took a touch with his chest and then, as he fell, hooked a sublime volley over Navas's despairing dive and beneath the crossbar. As furniture goes, Zidane is an exquisitely designed trophy cabinet.
"The truth is it has been a spectacular season", added Ronaldo.
Modric sped to the byline on the right and crossed and Ronaldo darted ahead of Bonucci at the near post to guide a shot past Buffon.
And like Real's performance on the pitch at the Principality Stadium as they tore the Italian champions apart with goals from Casemiro and Marco Asensio adding to Ronaldo's double, there was no sign of the Madrid fans' tiring of success.