Diego Godin put Atletico ahead in the 36th minute and it took an injury time equalizer from Sergio Ramos to rescue Real and force the extra half hour.
Atletico held out until the second half of the added time until Gareth Bale headed Real ahead after superb work by Angel Di Maria.
Marcelo and Cristiano Ronaldo, with a penalty, his record 17th goal in the Champions League this season, rubbed salt into the wound for Atletico, who had been bidding for a La Liga/Champions League double.
Coach Diego Simeone, unhappy at the added minutes in regulation play, summed up their frustration as he ran on to the pitch to confront Real’s Raphael Varane after the final goal and had to be restrained by stewards.
It was a sorry end to a magnificent season for Los Rojiblancos, but ultimately the quality of Real, epitomized by the world’s most expensive player Bale, which prevailed.
It was the Welshman’s 22nd goal of the season and sixth in the Champions League but surely the most important of his career.
“This is what every footballer dreams of and it doesn’t come any bigger than that,” the Welshman told Sky Sports after his side’s triumph.